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Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity
Accepted Manuscript Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity N. Kolkovska, M. Dimova, N. Kutev PII: DOI: Reference:
S0378-4754(16)30027-1 http://dx.doi.org/10.1016/j.matcom.2016.03.010 MATCOM 4308
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Mathematics and Computers in Simulation
Received date: 1 October 2015 Revised date: 17 February 2016 Accepted date: 16 March 2016 Please cite this article as: N. Kolkovska, M. Dimova, N. Kutev, Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity, Math. Comput. Simulation (2016), http://dx.doi.org/10.1016/j.matcom.2016.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity N. Kolkovskaa,∗, M. Dimovaa, N. Kuteva a Institute
of Mathematics and Informatics of BAS, acad. G. Bonchev str., bl.8, 1113 Sofia, Bulgaria
Abstract The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed. Grillakis, Shatah and Strauss’ stability theory is applied for the investigation of the orbital stability or instability of solitary waves to the double dispersion equation. An analytical formula, related to some conservation laws of the problem, is derived. As a consequence, the dependence of orbital stability or instability on the parameters of the problem is demonstrated. A complete characterization of the values of the wave velocity, for which the solitary waves to the generalized Boussinesq equation are orbitally stable or unstable, is given. In the special case of a quadratic nonlinearity our results are reduced to those known in the literature. Keywords: stability, solitary waves, double dispersion equation
1. Introduction The aim of this paper is to investigate the orbital stability or instability of the solitary waves to the double dispersion equation utt − u xx − h2 uttxx + h1 u xxxx + f (u) xx = 0, 2
3
f (u) = au + bu ,
x ∈ R, t ∈ R+ ,
(1) (2)
with constant coefficients a, b, h1 , h2 , h1 > 0, h2 > 0 and initial data 1
u(0, x) = u0 (x), ut (0, x) = u1 (x); u0 (x) ∈ H 1 (R), u1 (x) ∈ L2 (R), (−∆)− 2 u1 (x) ∈ L2 (R).
(3)
Here (−∆)−s u = F −1 (|ξ|−2s F(u)), for s > 0 and F(u), F −1 (u) are the Fourier and the inverse Fourier transform. By a solitary wave we mean a travelling with velocity c wave of the form ϕc (x − ct), vanishing at infinity. Let us recall, that a solitary wave ϕc (x − ct) is said to be orbitally stable if any solution u(t, x) of (1) – (3) with initial data close to the solitary wave for t = 0 stays close to ϕc (x − ct) for every t > 0. ∗ Corresponding
author Email addresses: [email protected] (N. Kolkovska), [email protected] (M. Dimova), [email protected] (N. Kutev) Preprint submitted to Mathematics and Computers in Simulation February 17, 2016
The orbital stability of the solitary waves to the generalized Boussinesq equation utt − u xx + h1 u xxxx + f (u) xx = 0, with a single nonlinearity
x ∈ R, t ∈ R+
f (u) = |u| p−1 u, p > 1
(4) (5)
2 and h1 = 1 is obtained by Bona and Sachs [2] for 1 < p < 5 and p−1 4 < c < 1. Later on, Liu [12] p−1 2 proves instability result for the same problem for c ≤ 4 and 1 < p < 5 or c2 < 1 and p ≥ 5. Strong instability to (4), (5), i.e. instability by means of blow up of the solutions, is obtained in p−1 . [13, 14] when 0 < c2 < 2(p+1) The stability of solitary waves to the generalized Boussinesq equation (4) with h1 = 1 and quadratic-cubic nonlinearity (2) is analytically studied in [5]. The authors prove that for some values of the constants a and b all solitary waves are stable (see Table 2). The double dispersion equation (1) with a single nonlinearity (5) is studied in [4]. The authors find conditions on c and the parameters a, b and p, for which the solitary waves are orbitally stable. Moreover, strong instability is proved for c2 < c20 , where the constant c0 is explicitly given. In the present paper we study the orbital stability or instability of the solitary waves to the double dispersion equation (1) with quadratic-cubic nonlinearity (2) and parameters, satisfying the conditions a > 0, b < 0, h1 > h2 . (6)
This choice is motivated by the investigations in [7, 8], where equation (4) with nonlinearity (2) has been proposed to be relevant to pulse propagation in biomembranes. Recently, the model (4), (2) is revised in [3] from the viewpoint of the solid mechanics. More precisely, a higher order term h2 uttxx with h2 - a small positive constant, h2 < h1 , is added to (4). With this term (4) transforms into the double dispersion equation (1). In both models [3, 7, 8], the constants in the equation are supposed to satisfy (6). For more details about the discussed models see [3, 7, 8] and the references therein. In this study we first derive explicitly all positive solitary waves to (1), (2). These waves are obtained in four regimes, depending on the relations between parameters a, b, h1 and h2 . The orbital stability or instability of solitary waves is studied by constructing of a function d(c), related to some conservation laws of (1) – (3) and applying the convexity or concavity criterion in [2, 6, 12]. By a stability or instability region we mean the set of wave velocities c, for which the solitary waves are orbitally stable or unstable. In Theorem 2 we obtain an analytical formula for d′′ (c). This formula allows one to check the orbital stability or instability of the solitary waves with velocity c for every choice of the constants a, b, h1 and h2 , satisfying (6). As a consequence, in Theorem 3 we give a complete description of the regions of orbital stability or instability of the solitary waves to generalized Boussinesq equation (4), (2). In this way we improve the results in [5] for all admissible values of the velocity c (see remark 4). For double dispersion equation (1), (2) the orbital stability or instability of the solitary waves is studied in neighbourhoods of the end points of the admissible values of the velocity c, see Theorem 5. In the special case of the generalized Boussinesq equation and double dispersion equation with a single quadratic nonlinearity our results reduce to the wellknown ones in the literature. The paper is organized in the following way. In Section 2 explicit formulas for the solitary waves to (1), (2) and (4), (2) are obtained. An analytical formula for d′′ (c), which determines 2
the orbital stability or instability of the solitary waves to (1), (2), is derived in Section 3. As an application of this formula, in Section 4 a complete investigation of the stability or instability of the solitary waves to (4), (2) is done, while Section 5 deals with an application to the double dispersion equation (1), (2). For simplicity, throughout the paper, for functions depending on x and t we use the short notation kuk = ku(·, t)kL2 (R) . By x+ we denote the first truncated power function: x+ = 0, if x ≤ 0, and x+ = x, if x > 0. 2. Solitary waves to double dispersion equation The solitary waves ϕc (x − ct) = ϕc (ζ) of the double dispersion equation (1), (2) with constant velocity c are solutions to the problem (h1 − h2 c2 )ϕ′′c (ζ) − (1 − c2 )ϕc (ζ) + aϕ2c (ζ) + bϕ3c (ζ) = 0, ϕc (ζ) → 0 for |ζ| → ∞.
(7)
When h2 = 0 the solutions to (7) are wellknown (see e.g. [11]) and are given in the following statement. Lemma 1. There exists a unique (up to translation of the coordinate system) positive solitary wave ϕc (ζ) to (4), (2) with constant velocity c −1 s r 1 − c2 9 ϕc (ζ) = 3(1 − c2 ) a + a2 + b(1 − c2 ) cosh ζ , ζ = x − ct, 2 h1
(8)
when one of the following assumptions is satisfied: (i) b > 0, (ii) b < 0,
a ∈ R, c2 ∈ [0, 1); a > 0, c2 ∈ 1 +
2a2 9b +
,1 .
The following lemma gives conditions on the parameters h1 , h2 , a and b for existence of positive solitary waves to (1), (2). Lemma 2. There exists a unique (up to translation of the coordinate system) positive solitary wave ϕc (ζ) of (1), (2) with constant velocity c −1 s r 2 1 − c 9 , ζ = x − ct, (9) 2 2 2 2 ζ ϕc (ζ) = 3|1 − c | a sgn(1 − c ) + a + b(1 − c ) cosh 2 h1 − h2 c2
when one of the following assumptions is fulfilled: h (i) b > 0, a ∈ R, c2 ∈ 0, min 1, hh12 ; 2 2 (ii) b > 0, a < 0, h1 < h2 1 + 2a c2 ∈ max 1, hh12 , 1 + 2a 9b , 9b ; h 2 2 1 c2 ∈ 1 + 2a (iii) b < 0, a > 0, h1 > h2 1 + 2a 9b + , 9b + , min 1, h2 ; (iv) b < 0, a ∈ R, c2 > max 1, hh21 . 3
Proof. We apply [1, Theorem 5], which gives necessary and sufficient conditions for existence of positive solutions to (7). According to [1, Remark 6.2.(ii)], the sign condition (h1 − h2 c2 )(1 − c2 ) > 0 necessarily holds. We look for positive solutions to (7) of the form r r h1 − h2 c2 h 1 − h 2 c2 = ψ(η), ζ = ϕc (ζ) = ϕc η η, 1 − c2 1 − c2
(10)
(11)
where ψ(η) satisfies the equation ψ′′ (η) − ψ(η) +
b a ψ2 (η) + ψ3 (η) = 0, ψ(η) → 0 for |η| → ∞. 2 1−c 1 − c2
(12)
From [10, Lemma 4.1] problem (12) has a unique (up to translation of the coordinate system) positive solution when either b > 0, a ∈ R (13) 1 − c2 or b 9b a a2 + (14) < 0, > 0, > 0. 2 2 2 1−c 1−c 2 1 − c2 1 − c2 This solution is given by the formula s −1 a 9b a2 + + ψ(η) = 3 cosh(η) 2 2 2 1−c 2 1−c 1 − c2 r −1 9b 2 2 2 2 1 − c cosh(η) . = 3|1 − c | a sgn(1 − c ) + a + 2
Using (11), we return back to function ϕc and obtain the form (9) of the unique positive solution to (7). Let us consider the restrictions on the parameters a, b, h1 , h2 and the velocity c, which guarantee the existence of solitary waves. Inequalities (10) and (13) are equivalent to
or
b > 0, 1 − c2 > 0, h1 − h2 c2 > 0, a ∈ R,
(15)
b < 0, 1 − c2 < 0, h1 − h2 c2 < 0, a ∈ R.
(16)
1 − c2 < 0, b > 0, a < 0, 2a2 + 9b(1 − c2 ) > 0, h1 − h2 c2 < 0
(17)
1 − c2 > 0, b < 0, a > 0, 2a2 + 9b(1 − c2 ) > 0, h1 − h2 c2 > 0.
(18)
Conditions (15) coincide with assumptions in Lemma 2(i), while conditions (16) - with assumptions in Lemma 2(iv). Analogously conditions (10) and (14) hold when
or
Conditions (17) lead to assumptions in Lemma 2(ii), while (18) is equivalent to assumptions in Lemma 2(iii). Lemma 2 is proved. 4
2
Remark 1. When condition (iii) of Lemma 2 holds and additionally 1 + 2a 9b < 0, then there exists a solitary wave ϕc with zero velocity c = 0, i.e. a ground state solution to (1), (2). The same observation is true when condition (ii) in Lemma 1 holds. Remark 2. For h1 = h2 = 1 the results in Lemma 2(i) and Lemma 2(iii) coincide with the results in [5, Theorem 1]. 3. Main results Throughout the rest of the paper we focus our investigation on the case ! ! 2a2 ,1 , b < 0, a > 0, h1 > h2 , c2 ∈ 1 + 9b +
(19)
i.e. we suppose that the conditions in Lemma 2(iii) are satisfied. We take into account, that problem (1), (2) with assumptions (19) is directly applicable to the model of pulse propagation in biomembranes, as it is mentioned in the Introduction. The other possible combinations of parameters, for which solitary waves to (1), (2) exist, i.e. the cases in Lemma 2(i), Lemma 2(ii), and Lemma 2(iv), will be treated in a forthcoming paper. In this section we formulate and prove the main results in the present paper. In order to study orbital stability or instability of the solitary waves ϕc (x − ct) we use the results from [2, 6, 12]. The proof is based on convexity or concavity of a function d(c) related to some conservation laws for equation (1). We rewrite problem (1), (2) to the following equivalent form ut = w x , wt = E − h2
∂2 ∂x2
!−1
E − h1
! ! ∂2 2 3 u − (au + bu ) x x , ∂x2
(20)
where E is the identity operator. The energy and momentum functionals of the system, given by ! Z u4 u3 1 dx, (21) u2 + h1 (u x )2 + w2 + h2 (w x )2 − 2a − b H(u, w) = 2 R 3 2 Z (uw + h2 u x w x ) dx, (22) M(u, w) = R
are conserved in time. We define also the functionals Ic (u) and Jc (u) as
Z au3 + bu4 dx, Ic (u) = h1 − h2 c2 ku x k2 + 1 − c2 kuk2 − R
! Z 1 au3 bu4 1 2 2 2 2 Jc (u) = dx. h1 − h2 c ku x k + 1 − c kuk − + 2 2 3 4 R
As in [2, 6, 12], we define the function d(c) (called ’the moment of instability’) as d(c) := H(ϕc , −cϕc ) + cM(ϕc , −cϕc ), with ϕc , H and M given by the expressions (9), (21) and (22) respectively. Now we formulate the following theorem: 5
Theorem 1. Suppose the parameters h1 , h2 , a and b are fixed and satisfy condition (19). Then d(c) = Jc (ϕc ),
(23)
where ϕc is the positive solitary wave to (1), (2) defined in (9). The function d(c) is given by the expression √ !1 1 4 h1 (1 − c2 ) 2 h2 2 2 d(c) = 1− c 9b2 h1 √ 2 p √ 2 2) 2a 2a + 9b(1 − c ) 2a + 3 −b(1 − c 2 log √ . (24) a + 3b(1 − c2 ) − p 1 1 12(−b) 2 (1 − c2 ) 2 2a − 3 −b(1 − c2 )
Proof. It is easy to check that the following identity
H(u, w) + cM(u, w) = 0.5kw + cuk2 + 0.5h2kw x + cu x k2 + Jc (u) holds. Since d(c) = H(ϕc , −cϕc ) + cM(ϕc , −cϕc ) = Jc (ϕc ),
then (23) is proved. We evaluate explicitly the function d(c). Since the solitary wave ϕc satisfies Ic (ϕc ) = 0, then Z Z 1 1 1 3 ϕ (x) dx + b ϕ4 (x) dx. (25) d(c) = Jc (ϕc ) = Jc (ϕc ) − Ic (ϕc ) = a 2 6 R c 4 R c q 1−c2 x we obtain from (25) the equality After the change of the variable s = exp h1 −h2 c2
Z ∞ s2 ds a d(c) =36(1 − c ) (h1 − h2 c ) 3 2 9 (a2 + b(1 − c2 )) 2 0 (s − 2νs + 1)3 2 Z ∞ 2 9b(1 − c ) s3 ds a + , where ν = − 1 . (26) 9 2 − 2νs + 1)4 2 2 2 (s (a + 2 b(1 − c )) 0 a2 + 92 b(1 − c2 ) 2 R ∞ sm−1 ds R∞ ds We evaluate integrals Pm = 0 (s2 −2νs+1) . m , m = 3, 4, by means of the integrals Qn = 0 (s2 −2νs+1)n Using the recurrence relation 2
5 2
2
1 2
2n − 3 ν + Qn−1 , n = 2, 3, 4, ... 2 2(n − 1)(1 − ν ) 2(n − 1)(1 − ν2 ) we get from (26) the expression d(c) as a function of Q1 : √ ! 12 !1 1 9 4 h1 (1 − c2 ) 2 h2 2 2 2 2 2 2 3b(1 − c ) + a − a a + b(1 − c ) Q1 . d(c) = 1− c 2 9b h1 2 Qn =
(27)
Under condition (19) we have ν2 =
√ 1 ν − ν2 − 1 a2 > 1 and Q = log . √ √ 1 a2 + 92 b(1 − c2 ) 2 ν2 − 1 ν + ν2 − 1
Finally, substituting Q1 in (27), we get the formula (24) for d(c). The proof of Theorem 1 is completed. 6
Let us consider the limiting case of (1), (2) with b = 0. In this case the double dispersion equation (1) transforms to problem with quadratic nonlinearity f (u) = au2 . As a consequence of Theorem 1 we have the following result. Corollary 1. Suppose the parameters of the double dispersion equation (1), (2) satisfy h1 > h2 , a > 0, b = 0 and c2 ∈ (0, 1). Then the function d(c) is reduced to the expression √ !1 6 h1 h2 2 2 2 25 (1 − c ) d(c) = 1 − . (28) c h1 5a2 Proof. In order to prove (28) we evaluate limb→0,b<0 d(c). We expand the logarithmic function in (24) in series around b = 0. We substitute the result into (24) and get formula (28). Let us note that (28) is identical with formula (4.1) from√[4] for p = 2 and a = 1. Indeed, for equation (1) with f (u) = au2 we have from [9] that d(0) = 65ah2 1 . In this way (28) transforms into formula (4.1) from [4]. According to the results in [2, 6, 12] the orbital stability or instability of the solitary waves to equation (1), (2) depends on the sign of the second derivative of the function d(c). In the following theorem we obtain an explicit formula of the second derivative d ′′ (c). Theorem 2. Suppose the parameters h1 , h2 , a and b are fixed and satisfy condition (19). Then the positive solitary wave ϕc of (1), (2) defined in (9) is orbitally stable if d′′ (c) > 0 and orbitally unstable if d ′′ (c) < 0, where p √ √ √ √ 2a + 3 −b 1 − c2 1 − c2 2a 4 h1 ′′ f (c) + √ g(c) log √ d (c) = p , 3 2 2 2 h2 2 2 2a + 9b − 9bc 12 −b 2a − 3 −b 1 − c 2 9b 1 − h1 c (29) ! h2 h2 h2 h2 2 h2 4 c a − 15a2b − 27b2 + 162b2 + 87a2 b + 378b2 h1 h1 h1 h1 h1 h22 h2 h2 4 2 2 2 2 h2 + −54a b 2 − 513b − 243b 2 c + 324b2 22 c6 , h1 h1 h1 h1
f (c) = − 18a2 b − 81b2 − 2
g(c) =18b + 2a2
h2 h2 h2 h2 + 9b − 81b c2 + 54b 22 c4 . h1 h1 h1 h1
Proof. Now we apply the result in [6] for orbital stability or instability of solitary waves to problem (20). The nonlinear term f (u) = au2 + bu3 satisfies all conditions of [12, Lemma 2.1] under assumptions (iii) of Lemma 2. Following the ideas in [2, 6, 12], one can prove that the stability or instability statements in [2, 6] holds also for the double dispersion equation (1), (2). Tedious calculations of derivatives give us from (24) formula (29). Theorem 2 is proved. Theorem 2 gives a direct method for studying orbital stability or instability of solitary waves ϕc of (1), (2). In order to check the stability or instability of the solitary waves one has only to substitute the values of the actual parameters a, b, h1 , h2 and c into formula (29). For the double dispersion equation (1) with a quadratic nonlinearity, as in Corollary 1, we evaluate the limit of d ′′ (c) for b → 0 with b < 0. 7
Corollary 2. Suppose the parameters of the double dispersion equation (1), (2) satisfy h1 > h2 , a > 0, b = 0 and c2 ∈ (0, 1). Then d ′′ (c) is given by the following formula d′′ (c) =
√ 1 6 h1 (1 − c2 ) 2 1 − 5a2 !2 h2 6 30 h1 c − 15
!− 3 h2 2 2 c h1 !2 ! h2 h2 4 h2 h2 2 − 5 + 51 c + 22 + 20 c − . h1 h1 h1 h1
(30)
4. Application to generalized Boussinesq equation
The investigation of the sign of d ′′ (c), i.e. the orbital stability or instability for the solitary waves to (1), (2) is a difficult problem because of numerous (exactly five) parameters. But, in the case of the generalized Boussinesq equation (4), (2) such examination can be done. In the following we set √ b k = 2 , ξ = 1 − c2 . a Then from Lemma 1(ii) and Remark 1 we have either ! 2 2 2 when k ≤ − , ξ ∈ 0, − 9k 9 or
2 < k < 0. 9 In the new variable ξ and for h2 = 0 formula (29) is rewritten as √ √ √ ! q 3(2 + 9k)ξ 2 2 + 3ξ −k 3a2 (−k) ′′ 2 =: G(k, ξ) 1 − ξ = 6ξ − − √ log √ √ √ d 2 + 9kξ2 4 h1 2 −k 2 − 3ξ −k ξ2 ∈ (0, 1] when −
(31)
In the following theorem we obtain explicit conditions on velocity c in terms of ab2 ∈ (−∞, 0) for which the solitary waves (8) are orbitally stable or unstable. We divide the admissible for ab2 interval (−∞, 0) into subintervals (−∞, k0 ), (k0 , − 29 ) and (− 92 , 0) because of the different behaviour of d′′ (c). The constant k0 is determined as the positive root of equation (37) and k0 ≈ −0.225397 < − 29 . Theorem 3. Suppose the parameters h1 , a and b are fixed and satisfy condition (19) with h2 = 0. If ϕc (x − ct) is the positive solitary wave to (4), (2), defined in (8), then the following statements hold for ϕc : 2
(i) If ab2 ∈ (−∞, k0 ), then all positive solitary waves ϕc with velocity c2 ∈ (1 + 2a 9b , 1) are orbitally stable. The value k0 is determined as the positive root of equation (37) and k0 ≈ −0.225397; 2
2 2 (ii) If ab2 ∈ (k0 , − 29 ), then all positive solitary waves ϕc with velocity c2 ∈ (1 + 2a 9b , c1 ) ∪ (c2 , 1) 2 2 2 are orbitally stable, while for c ∈ (c1 , c2 ) are orbitally unstable. Here the constants c1 , c2 are defined in (38);
8
(iii) If ab2 ∈ (− 92 , 0), then all positive solitary waves ϕc with velocity c2 ∈ (c23 , 1) are orbitally h stable, while for c2 ∈ 0, c23 are orbitally unstable. The constant c3 is defined in (41);
(iv) If ab2 = − 92 , then all positive solitary waves ϕc with velocity c2 ∈ (0, c24 ) are orbitally unstable, while for c2 ∈ (c24 , 1) are orbitally stable. The value of c4 is defined in (42) and c24 ∈ (0, 12 ).
Proof. According to Theorem 2, the sign of d′′ (c) determines the orbital stability or instability of solitary waves ϕc . Since the sign of d′′ (c) coincides with the sign of the defined in (31) function G(k, ξ), we study the behavior of the function G(k, ξ). Straightforward computations give us h i 27k ∂G (k, ξ) = 18kξ4 + (8 + 9k)ξ2 − 2 , 2 2 ∂ξ (2 + 9kξ ) h i 54kξ ∂2 G (k, ξ) = −81k2 ξ2 + 54k + 16 . 2 2 3 ∂ξ (2 + 9kξ )
(32)
From (32) it follows that
∂2 G 8 (k, ξ) > 0 for k < − and every ξ ≥ 0. ∂ξ2 27 Moreover, the equation
∂G ∂ξ (k, ξ)
(33)
= 0, considered as a polynomial of ξ, has real roots
p 1 (34) 8 + 9k ± (8 + 9k)2 + 144k 36k √ √ when either k ≤ − 89 2 + 3 or − 89 2 − 3 ≤ k < 0. 2 Case: k < − 92 , ξ2 ∈ 0, − 9k √ √ 8 8 , − 27 ≤ k ≤ − 98 2 − 3 and − 89 2 − 3 < This case is reduced to three subcases: k < − 27 k < − 29 . ξ±2 (k) = −
8 < k0 (35) 27 27 8 ∂G Since G(k, 0) = 0, ∂G ∂ξ (k, 0) = − 2 k > 0, from (33) we get for k < − 27 that ∂ξ (k, ξ) > 0, 8 G(k, ξ) > 0 for every ξ > 0 and hence d′′ (c) > 0. Thus for k < − 27 statement (i) of Theorem 3 follows from Theorem 2. Subcase: k < −
√ 8 8 ≤ k ≤ − 2 − 3 < k0 27 9 √ 40 Using the estimate 0 < 3 ≤ 9k + 16 ≤ 8 3, we obtain !2 2 ∂G 27 192 − (9k + 16) 8 + 9k 2 (k, ξ) = + ≥ 0 18 kξ + ∂ξ (2 + 9kξ2 )2 36 72 Subcase: −
(36)
2 ) and equality holds only for ξ2 = 8+9k for every ξ2 ∈ (0, − 9k 36k . Thus G(ξ, k) > 0 for every 2 2 ξ ∈ (0, − 9k ), when k is in the interval (36). Statement (i) is proved for k satisfying (36). 9
G(k0 , ξ)
G(k, ξ)
ξ1 (k) 0
•
•
•
•
ξ+ (k)
ξ− (k)
ξ2 (k) •
1
ξ •
0
(a) Figure 1: Function G(k, ξ), k =
•
ξ− (k0 )
ξ+ (k0 )
•
1
ξ
(b) b , a2
ξ=
√
√ 1 − c2 , for: (a) k ∈ − 89 (2 − 3), − 29 ; (b) k = k0 , k0 ≈ −0.225397.
√ 2 8 2− 3
i 54kξ± (k) h ∂2 G (k, ξ± (k)) = −81k2 ξ±2 (k) + 54k + 16 2 2 3 ∂ξ (2 + 9kξ± (k)) p p 27kξ± (k) (8 + 9k)2 + 144k (8 + 9k)2 + 144k ± 9k = 2 3 2(2 + 9kξ± (k))
and k < − 29 , it follows that ∂2 G (k, ξ− (k)) <0, ∂ξ2
p −1 432kξ+ (k) (8 + 9k)2 + 144k p ∂2 G (k, ξ+ (k)) = (8 + 9k)2 + 144k − 9k (2 + 9k) > 0. 2 2 3 ∂ξ (2 + 9kξ+ (k))
Hence ξ− (k) is a point of local maximum of G(k, ξ), while ξ+ (k) is a point of local minimum for G(k, ξ). q
2 2 From (31) we have G(k, 0) = 0, ∂G ∂ξ (k, 0) > 0 and from (2+9kξ )G(k, ξ) → −(2+9k) − k > 0 q as ξ → − 9k2 we conclude that ξ− (k) is a point of positive local maximum, while ξ+ (k) is a point 2 of local minimum for G(k, ξ) for ξ2 ∈ 0, − 9k . The value of the local minimum G(k, ξ+ (k)) can be a positive, negative or equal to zero number. Analysing the sign of G(k, ξ+ (k)) we conclude that there exists a constant k0 , which is a unique root of the equation
G(k, ξ+ (k)) = 0, where ξ+2 (k) = −
p 1 8 + 9k + (8 + 9k)2 + 144k , 36k
i.e. G(k0 , ξ+ (k0 )) = 0 (see Fig. 1(b)). Numerically we find out that k0 ≈ −0.225397. Hence, it follows that: 10
(37)
G(- 92 , ξ)
G(k, ξ)
•
0
ξ− (k)
1
•
•
ξ0 (k)
å
0
ξ
(a)
2 2
•
ξ0
1
•
ξ
(b)
Figure 2: Function G(k, ξ), k =
b , a2
ξ=
√
1 − c2 , for: (a) k ∈ − 92 , 0 ; (b) k = − 29 .
√ 2 • if − 98 (2 − 3) < k < k0 , then G(k, ξ) > 0 for every ξ2 ∈ (0, − 9k ) and the statement (i) is proved for this interval for k. Combining this result with the results in the previous subcases (35) and (36), we obtain the proof of statement (i) of Theorem 3. • if k0 < k < − 92 , then there exist points ξ1 (k), ξ2 (k), such that G(k, ξ1 (k)) = 0 and G(k, ξ2 (k)) = 0. 2 2 , G(k, ξ) > 0 for ξ ∈ (0, ξ1 (k)) ∪ (ξ2 (k), − 9k ) Moreover, ξ− (k) < ξ1 (k) < ξ+ (k) < ξ2 (k) < − 9k and G(k, ξ) < 0 for ξ ∈ (ξ1 (k), ξ2 (k)). (see Fig. 1(a).)
From the identity ξ2 = 1 − c2 and the sign of G(k, ξ), i.e. the sign of d′′ (c), statement (ii) is proved with constants c1 and c2 defined as c22 = 1 − ξ12 (k), Case: −
c21 = 1 − ξ22 (k).
(38)
2 < k < 0, ξ2 ∈ (0, 1] 9
In this case the function ∂G ∂ξ (k, ξ) has two real roots given in (34). Straightforward computations show that ξ+2 (k) > 1. Moreover, ξ−2 (k) = 1 −
p p −1 1 8 + 45k − (8 + 9k)2 + 144k = 1 − 6(2 + 9k) 8 + 45k + (8 + 9k)2 + 144k 36k
8 8 and for − 92 < k ≤ − 45 from the first equality and for − 45 < k < 0 from the second equality we 2 get ξ− (k) < 1. Since from (32) we have
h i ∂2 G 54kξ 54(−k)ξ 2 (k, ξ) = −81k2 ξ2 + 54k + 16 ≤ 4ξ + 12 − 16 2 2 3 2 3 ∂ξ (2 + 9kξ ) (2 + 9kξ ) 216(−k)ξ(ξ2 − 1) <0 = (2 + 9kξ2 )3 for every ξ ∈ (0, 1), it follows that G(k, ξ) is a strictly concave function w.r.t. ξ ∈ (0, 1). Thus ∂G ∂G 27 ∂ξ (k, ξ) has only one zero at ξ− (k) ∈ (0, 1) and from ∂ξ (k, 0) = − 2 k > 0 we get that ∂G ∂G (k, ξ− (k)) = 0; (k, ξ) > 0 for ξ ∈ (0, ξ− (k)); ∂ξ ∂ξ 11
∂G (k, ξ) < 0 for ξ ∈ (ξ− (k), 1]. ∂ξ
(39)
Tedious computations give us √
√ 2 + 3 −k G(k, 1) =3 − √ log √ √ <0 2 −k 2 − 3 −k 2
√
for k ∈ (− 29 , 0). Since G(k, 0) = 0 we get from (39) and G(k, 1) < 0 that there exists a unique point ξ0 (k), ξ02 (k) ∈ (ξ−2 (k), 1) such that G(k, ξ0 (k)) = 0; G(k, ξ) > 0 for ξ ∈ (0, ξ0 (k)); G(k, ξ) < 0 for ξ ∈ (ξ0 (k), 1]
(40)
(see Fig. 2(a)). The value of ξ0 (k) is defined from the equation √ √ √ 3(2 + 9k)ξ0 (k) 2 2 + 3ξ0 (k) −k log G(k, ξ0 (k)) = 6ξ0 (k) − − √ √ √ = 0. 2 + 9kξ02 (k) 2 −k 2 − 3ξ0 (k) −k For
c23 = 1 − ξ02 (k)
(41)
and (40) the proof of statement (iii) of Theorem 3 is completed. 2 Case: k = − , ξ2 ∈ (0, 1) 9 Now we investigate the special case k = − 92 . We have ! 2 3 1+ξ G − , ξ = 6ξ − log . 9 2 1−ξ 6ξ 2 3 ∂2 G 2 Since ∂G follows that G − 29 , ξ is a concave ∂ξ − 9 , ξ = 6 − 1−ξ2 and ∂ξ2 − 9 , ξ = − (1−ξ2 )2 < 0, it √ function, which has a positive maximum at the point ξ = 22 and limξ→1 G − 92 , ξ = −∞ (see √ Fig. 2(b)). Hence there exists a unique root ξ0 , ξ0 ∈ 22 , 1 , of the equation
1+ξ = 0, 1−ξ such that G − 92 , ξ > 0 for ξ ∈ (0, ξ0 ) and G − 92 , ξ < 0 for ξ ∈ (ξ0 , 1). Thus statement (iv) of Theorem 3 follows from Theorem 2 with constant c4 defined from 4ξ − log
c24 = 1 − ξ02 .
(42)
The proof of Theorem 3 is completed. Remark 3. If we substitute h2 = 0 in (30), we get the following expression for d′′ (c) √ 1 6 h1 lim d′′ (c) = (1 − c2 ) 2 20c2 − 5 2 b→0,b<0 5a
for the generalized Boussinesq equation (4) with quadratic nonlinearity f (u) = au2 . Since d′′ (c) < 0 for c2 < 41 and d′′ (c) > 0 for c2 > 14 , then, using Theorem 2, we establish the wellknown results for orbital stability or instability of solitary waves proved in [2, 12]. 12
Table 1: Regions of stability or instability of solitary waves to generalized Boussinesq equation (1) according to Theorem 3. The constants c1 , c2 , c3 , c4 and k0 are defined in the proof of Theorem 3 and k0 ≈ −0.225397.
Th. 3 b ∈ a2 c2 ∈
(i) (−∞, k0 ) 2
2
(ii) (k0 , − 92 )
(iv) − 92
(iii) (− 92 , 0)
2a 2 (1 + 2a (c21 , c22 ) (c22 , 1) (0, c24 ) (c24 , 1) [0, c23 ) (c23 , 1) 9b , 1) (1 + 9b , c1 ) stable stable unstable stable unstable stable unstable stable
Let us recall the result from [5] for orbital stability or instability of the solitary waves to problem (4), (2). The statement is suitably altered to fit the present context. Theorem 4 ([5]). Suppose b < 0, a > 0 are constants and h1 = 1.
(i) If ab2 ∈ (−∞, − 31 ], then all positive solitary waves of (4), (2) with velocity c2 ∈ 1 + are orbitally stable;
2a2 9b
,1
(ii) If ab2 ∈ (− 29 , 0), then the positive solitary waves of (4), (2) are orbitally stable for c2 ∈ (c2♯ , 1) and orbitally unstable for c2 ∈ (0, c2♭ ), where c♯ and c♭ are some constants satisfying the inequalities 0 < c♭ < c♯ < 1. Remark 4. Let us compare the results in Theorem 3 with the corresponding ones in Theorem 4. For convenience, we illustrate the results in Theorem 3 and Theorem 4 by means of Table 1 and Table 2, correspondingly. • the case
b a2
∈ (− 31 , − 29 ] is studied in Theorem 3 for the first time.
• in case ab2 ∈ (− 92 , 0) the results in Theorem 3 improve the result in [5]. More precisely, in [5] the orbital stability or instability is investigated only in a small neighbourhood of c = 0 and c = 1, while in Theorem 3 the result is extended to the whole interval [0, 1) for c. Remark 5. The results for stability or instability of solitary waves are very sensitive with respect to the ratio ab2 for ab2 ∈ (− 31 , 0). For example when ab2 ∈ (k0 , − 92 ], i.e. an interval with length less than 0.0032, the stability or instability of solitary waves is quite different in comparison with the corresponding one in the neighbouring intervals (− 31 , k0 ) and (− 92 , 0). 5. Application to double dispersion equation In this section we apply Theorem 2 to double dispersion equation (1), (2). The results are not so comprehensive as in Theorem 3. They outline the influence of the ratio of the dispersive coefficients hh12 for orbital stability or instability of the solitary waves in neighbourhoods of the end points of the admissible values of c. Theorem 5. Suppose the parameters h1 , h2 , a and b are fixed and satisfy condition (19). If ϕc (x − ct) is the positive solitary wave of (1), (2), defined in (9), then the following assertions hold for ϕc : 13
Table 2: Regions of stability or instability of solitary waves to generalized Boussinesq equation (1) according to Theorem 4 in [5]. The constants c♭ , c♯ are defined in the proof of Theorem 4 and k0 is defined in the proof of Theorem 3, k0 ≈ −0.225397.
Th. 4([5]) b ∈ a2 c2 ∈
(i) (−∞, − 31 ]
(1 +
2a2 9b , 1)
stable
(− 31 , k0 )
(k0 , − 29 ]
2
2
(1 + 2a 9b , 1) not considered
(1 + 2a 9b , 1) not considered
(ii) (− 92 , 0) (0, c2♭ ) unstable
[c2♭ , c2♯ ] not considered
(c2♭ , 1) stable
(i) in a neighbourhood of the right-end point c = 1:
2 , 1 , such that for for every ab2 ∈ (−∞, 0) there exists a constant C1 , C12 ∈ 1 + 2a 9b + c2 ∈ C12 , 1 all positive solitary waves ϕc with velocity c are orbitally stable;
(ii) in a neighbourhood of the left-end point c = 0 or c = 1 + (ii)a if
b a2
2a2 9b
:
∈ (−∞, − 92 ), then there exists a constant C2 , C22 ∈ 1 + 2
2a2 9b , 1
, such that for
2 c2 ∈ (1 + 2a 9b , C2 ) all positive solitary waves ϕc with velocity c are orbitally stable; (ii)b if ab2 ∈ − 29 , 0 , then there exists a constant C3 ab2 , hh12 , C32 ∈ (0, 1), such that all positive solitary waves ϕc with velocity c are orbitally unstable for c2 ∈ (0, C32 );
(ii)c if ab2 = − 29 , then there exists a constant C4 , such that all positive solitary waves ϕc with velocity c2 ∈ (0, C42 ) are orbitally unstable.
The constants Ci , i = 1, 2, 3, 4, depend on
b a2
and
h2 h1 .
Proof. We use the short notations √ h2 , ξ = 1 − c2 . h1 Note that µ ∈ (0, 1) and for k ≤ − 92 we get ξ2 ∈ 0, − 9k2 , while for k ∈ (− 29 , 0) we have ξ2 ∈ (0, 1]. Using the new notations, we rewrite formula (29) as √ √ ! 3 5 q 2 + 3ξ −k 9a2 (−k) 2 (1 − µc2 ) 2 ′′ 2 d 1 − ξ = G(k, ξ, µ) = G1 (k, ξ, µ) log √ √ √ + G2 (k, ξ, µ), 4 h1 2 − 3ξ −k (43) k=
with
b , a2
µ=
√
2 18k + 2µ(1 − 36k) + 54kµ2 + 27kµ(3 − 4µ)ξ2 + 54kµ2 ξ4 , 12 2 + 9kξ2 = µ2 81k2 − 54k + (108k − 486k2 )ξ2 + (729k2 − 54k)ξ4 − 324k2 ξ6 G2 (k, ξ, µ) √ −kξ + µ −162k2 + 72k − 2 + (648k2 − 87k)ξ2 − 513k2 ξ4 + 81k2 − 18k − 162k2 ξ2 ,
G1 (k, ξ, µ) =
14
Proof of (i) Simple computations lead to 5
G(k, 0, µ) = 0,
∂G 81(−k) 2 (k, 0, µ) = (1 − µ)2 > 0 ∂ξ 2
and from (43), (44) we get that d′′ (1) > 0. Hence, there exists a constant C1 , C12 ∈ 1 + such that d′′ (c) > 0 for c2 ∈ C12 , 1 This proves assertion (i) of Theorem 5. Proof of (ii)a If k = Since
b a2
√
√ 2 + 3ξ −k =0 lim √ √ 2 (2 + 9kξ ) log √ 2 − 3ξ −k ξ→ − 9k
2a2 9b +
< − 92 , we calculate the limit of (2 + 9kξ2 )G(k, ξ, µ) as ξ → 2
it follows that
(44) ,1 ,
q 2 . − 9k
2 lim √ 2 (2 + 9kξ )G(k, ξ, µ) =
ξ→
− 9k
√ ! 2 2 8 2 2 2 µ (81k + 54k + 12 + ) + µ(−162k − 72k − 8) + 81k + 18k) 3 9k √ ! µ2 2 (9k + 2) 9(1 − µ)2 k − 4µ(1 − µ) + >0 = 3 9k
for k = ab2 < − 29 and µ ∈ (0, 1). 2 Hence, there exists a constant ξ2 , 0 < ξ22 < − 9k2 such that G(k, ξ, µ) > 0 for ξ2 ∈ (ξ22 , − 9k ). If 2 2 ′′ 2 2 C2 = 1 − ξ2 , then we get d (c) > 0 for c ∈ (0, C2 ) and statement (ii)a in Theorem 5 is proved. Proof of (ii)b In order to prove (ii)b we calculate √ √ √ 5 √ 9(−k) 2 ′′ 2 2 + 3 −k (18k + 2µ + 9kµ) log √ √ d (0) = − −k(9k + 3kµ + µ) + √ = G(k, 1, µ). 12 4 h1 2 − 3 −k We evaluate also the derivatives √ ∂G 1 (k, 1, µ) = √ (9k + 3kµ + µ) − −k(9 + 3µ) ∂k 2 −k √ √ √ 2 2 + 3 −k 18k + 2µ + 9kµ + (6 + 3µ) log √ √ − √ 4 2 − 3 −k 2 −k(2 + 9k) √ √ √ √ √ 3 2 2 + 3 −k 9 9 −k (2 + µ) log √ , = √ − (3 + µ) −k + 4 2 + 9k 2 − 3 −k 2
√ ∂2 G 9(3 + µ) 9 81 −k 9(2 + µ) + − − (k, 1, µ) = − √ √ √ ∂k2 −k(2 + 9k) 4 −k 2 −k(2 + 9k) (2 + 9k)2 9 = √ −4(2 + µ)(2 + 9k) + (3 + µ)(2 + 9k)2 − 2(2 + 9k) + 36k 4 −k(2 + 9k)2 18 ((2 + 9k)(27k + 9kµ − 2µ) + 36k) < 0. = √ 4 −k(2 + 9k)2 15
b a2
d′′ (c)
= −0.23
d′′ (c)
h2 h1
h2 h1 =0.9
= 0.1
h2 h1 =0.9 h2 h1
= 0.1
0
1
|
0
0.1839 ≈
q
1+
2a2 9b
b a2
1 c
(a) Figure 3: Function d′′ (c) for different values of
c
= −0.21
(b) h2 h1
and: (a) k =
b a2
= −0.23 < − 92 ; (b) k =
b a2
= −0.21 > − 92 .
Hence G(k, 1, µ) is a strictly concave function w.r.t. k for k ∈ (− 92 , 0). Moreover, there exists a 2 constant ξ3 such that ∂∂kG2 (k, ξ, µ) < 0 for ξ2 ∈ (ξ32 , 1] and k ∈ (− 92 , 0). From ∂G ∂k (0, 1, µ) = 0 it 2 2 (k, ξ, µ) > 0 for ξ ∈ (ξ , 1]. follows that ∂G 3 ∂k Thus G(k, ξ, µ) is a strictly increasing function w.r.t. k for ξ2 ∈ (ξ32 , 1]. From G(0, 1, µ) = 0 we get G(k, ξ, µ) < 0 for ξ2 ∈ (ξ32 , 1], i.e. we have d ′′ (c) < 0 for k ∈ (− 92 , 0), c2 ∈ (0, C32 ) with C3 defined as C32 = 1 − ξ32 . In this way we proved statement (ii)b in Theorem 5. Proof of (ii)c Let k = − 92 . In order to evaluate limξ→1 G(− 92 , ξ, µ) we calculate functions G1 and G2 , defined in (43), for k = − 29 , i.e. ! √ ! 2ξ 2 2 2 2 2 2 µ 16(1 − ξ ) + µ(38ξ − 39) + 8 , G2 − , ξ, µ = 9 6 3 √ ! 2 2 (45) 1 + 3µ2 − 6µ2 ξ2 + 3µ2 ξ4 . G1 − , ξ, µ = − 9 3 √ √ (12−µ) 2 2 2 . Hence lim G − , ξ, µ = − We get from (45) that limξ→1 G2 − 29 , ξ, µ = (12−µ) ξ→1 9 9 √ 9 1+ξ 2 2 From the continuity of G − 9 , ξ, µ w.r.t ξ, there exists ξ4 3 limξ→1 log 1−ξ = −∞ for µ ∈ (0, 1). 2 2 such that for ξ ∈ (ξ4 , 1] we have G − 92 , ξ, µ < 0. Applying Theorem 2, we get statement (ii)c of Theorem 5 for c2 ∈ (0, C42 ) and C4 defined from C42 = 1 − ξ42 . Theorem 5 is completely proved. Let us consider once again the limiting case b = 0, i.e. the double dispersion equation (1) with quadratic nonlinearity f (u) = au2 . Lemma 3. Suppose the parameters of the double dispersion equation (1), (2) satisfy h1 > h2 , a > 0, b = 0 and c2 ∈ (0, 1). Then there exists a constant c∗ , c2∗ ∈ (0, 1) such that all positive solitary waves ϕc to (1), (2) with velocity c2 ∈ (0, c2∗ ) are orbitally unstable, while with velocity c2 ∈ (c2∗ , 1) are orbitally stable. Proof. According to (30), the sign of d′′ (c) in the interval c2 ∈ (0, 1) depends only on the sign of the polynomial F(c2 ) !2 !2 ! h2 h2 h h2 h 2 2 2 6 F(c ) = 30 − 5. c − 15 + 51 c4 + 22 + 20 c2 − h1 h1 h1 h1 h1 16
d′′ (c)
- region of stability - region of instability
|
0
Figure 4: Function d′′ (c) and the regions of stability or instability of solitary waves for q 2 1 + 2a 9b ≈ 0.1007.
Let η = c2 and µ =
h2 h1 .
1
0.1007
h2 h1
c
= 0.9, k =
b a2
= −0.2245 and
Using these notations, we rewrite F as a new function F1 ,
F1 (η) := F(c2 ) = 30µ2 η3 − 15µ2 + 51µ η2 + (22µ + 20) η − µ − 5,
which is a third order polynomial of η. From the Sturm’s theorem for the number of the real roots of the equation F1 (η) = 0, (46) it follows that (46) has only one positive root in the interval η ∈ (0, 1). Indeed, the number of variations W(0) of the signs of the sequence F1 (0) = −µ − 5 < 0, F1′ (0) = 22µ + 20 > 0, F1′′ (0) = −30µ2 − 102µ < 0, F1′′′ (0) = 180µ2 > 0 is 3, i.e. W(0) = 3, for every µ ∈ (0, 1). Correspondingly, W(1) = 2 for the sequence F1 (1) = 15(µ − 1)2 > 0, F1′ (1) = 20(1 − µ)(1 − 3µ), F1′′ (1) = 6µ(25µ − 17), F1′′′ (1) = 180µ2 > 0, 17 17 calculated for each admissible value of µ: µ ∈ (0, 13 ), µ = 31 , µ ∈ ( 13 , 17 25 ), µ = 25 , µ ∈ ( 25 , 1) and neglecting the zero terms. Thus equation (46) has one real root η∗ in the interval (0, 1) because W(0) − W(1) = 1. Since F1 (0) is negative, we have for η < η∗ that F1 (η) < 0, while for η > η∗ we have F1 (η) > 0. The rest of the proof follows from Theorem 2 and Lemma 3 is proved for c2∗ = η∗ .
Note that the result in Lemma 3 is announced and numerically demonstrated in [4]. The analytical investigation of the regions of stability or instability of the solitary waves to (1), (2) is a quite complicated problem as it is clear from the proof of Theorem 5. However, for every particular choice of the parameters of (1), (2) the application of Theorem 2 simplifies this problem. Thus the regions of stability or instability of the solitary waves are obtained for every admissible choice of c. On Fig. 3 for two special choices of the quantities ab2 and hh12 the graphs of d′′ (c) are given. Figure 3(a) illustrates the statement of Theorem 5(i) and Theorem 5(ii)a , while Figure 3(b) illustrates Theorem 5(i) and Theorem 5(ii)b . 17
On Fig. 4 the graph of d′′ (c) for parameters hh12 = 0.9 and k = ab2 = −0.2245 is presented. One can observe that the graph of d′′ (c) is completely different from the graphs of d′′ (c) for values of k close to −0.2245 and fixed ratio hh21 = 0.9: • for k = −0.23 all solitary waves are stable, see Fig. 3(a); • for k = −0.21 the admissible region (0, 1) of the velocity c2 is divided into two subintervals. The first one is a region of instability and the second one - a region of stability, see Fig. 3(b); • for k = −0.2245 we have two regions of stability, divided by a region of instability, see Fig. 4. 6. Conclusions In this paper the orbital stability or instability of solitary waves to double dispersion equation (1) and generalized Boussinesq equation (4) with quadratic-cubic nonlinearity f (u) = au2 + bu3 , a > 0, b < 0 is studied. All positive solitary waves to the double dispersion equation are explicitly found. An analytical formula for d ′′ (c) is derived for double dispersion equation (1), (2). This formula allows one to obtain the regions of stability or instability for every particular choice of the parameters of problem (see Theorem 2). In the special case of generalized Boussinesq equation (4) the orbital stability or instability of the solitary waves is completely investigated for all admissible velocities c (see Theorem 3). In particular, the case ab2 ∈ (− 31 , − 92 ) is investigated for the first time in this study. For double dispersion equation (1), (2) the orbital stability or instability of the solitary waves is studied in neighbourhoods of the end points of the admissible values of the velocity c (see Theorem 5). The statements depend only on the ratios hh12 and ab2 . For h1 > h2 , the results do not essentially affected by the values of h2 . In the limiting case h2 = 0 and b = 0 the statement in Theorem 3 reduces to results in [2, 12]. When h2 , 0 and b = 0, Theorem 5 coincides with the result in [4], see Corollary 3. Acknowledgement This research is partially supported by the Bulgarian Science Fund under grant DFNI I-02/9. References [1] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of solitary waves, Arch. Rat. Mech. Anal., 82 (1983) 313–346. [2] J. Bona, R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys., 118 (1988) 15–19. [3] J. Engelbrecht, K. Tamm, T. Peets, On mathematical modelling of solitary pulses in cylindrical biomembranes, Biomech. Model. Mechanobiol., 14 (2015) 159–167. [4] H.A. Erbay, S. Erbay, A. Erkip, Instability and stability properties of traveling waves for the double dispersion equation, Nonlinear Analysis, 133 (2016) 1–14 [5] H. Freist¨uhler, J. H¨owing, An analytical proof for the stability of Heimburg-Jackson pulses, ArXiv:1303.5941v1 (2013) [6] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, Journal of Functional Analysis, 74 (1987) 160–197. [7] T. Heimburg, A.D. Jackson, On soliton propagation in biomembranes and nerves, Proc. Natl. Acad. Sci USA, 102 (2005), 9790–9795.
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[8] T. Heimburg, A.D. Jackson, On the action potential as a propagating density pulse and the role of anesthetics, 57 (2007) Biophys. Rev. Lett., 2 (2007) 57–78. [9] N. Kutev, N. Kolkovska, M. Dimova, Theoretical and numerical aspects for global existence and blow up for the solutions to Boussinesq paradigm equation, AIP Conf. Proc. 1404 (2011) 68–76. [10] N. Kutev, N. Kolkovska, M. Dimova, Global existence to generalized Boussinesq equation with combined powertype nonlinearities, J. Math. Anal. Appl. 410 (2014) 427-444. [11] B. Lautrup, R. Appali, A. Jackson, T. Heimburg, The stability of solitons in biomembranes and nerves, Eur. Phys. J. E, 34 (2011) 57–65 [12] Y. Liu, Instability of solitary waves for generalized Boussinesq equation, J Dynamics Differ. Equat., 5 (1993) 537–558. [13] Y. Liu, Instability and blow-up of the solutions to a generalized Boussinesq equation, SIAM J Math. Anal., 26 (1995) 1527–1546. [14] Y. Liu, M. Ohta, G. Todorova, Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations, Annales de l’Inst. Henri Poincare (c) Non-linear Analysis, 24 (2007) 539 –549.
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S0378-4754(16)30027-1 http://dx.doi.org/10.1016/j.matcom.2016.03.010 MATCOM 4308
To appear in:
Mathematics and Computers in Simulation
Received date: 1 October 2015 Revised date: 17 February 2016 Accepted date: 16 March 2016 Please cite this article as: N. Kolkovska, M. Dimova, N. Kutev, Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity, Math. Comput. Simulation (2016), http://dx.doi.org/10.1016/j.matcom.2016.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity N. Kolkovskaa,∗, M. Dimovaa, N. Kuteva a Institute
of Mathematics and Informatics of BAS, acad. G. Bonchev str., bl.8, 1113 Sofia, Bulgaria
Abstract The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed. Grillakis, Shatah and Strauss’ stability theory is applied for the investigation of the orbital stability or instability of solitary waves to the double dispersion equation. An analytical formula, related to some conservation laws of the problem, is derived. As a consequence, the dependence of orbital stability or instability on the parameters of the problem is demonstrated. A complete characterization of the values of the wave velocity, for which the solitary waves to the generalized Boussinesq equation are orbitally stable or unstable, is given. In the special case of a quadratic nonlinearity our results are reduced to those known in the literature. Keywords: stability, solitary waves, double dispersion equation
1. Introduction The aim of this paper is to investigate the orbital stability or instability of the solitary waves to the double dispersion equation utt − u xx − h2 uttxx + h1 u xxxx + f (u) xx = 0, 2
3
f (u) = au + bu ,
x ∈ R, t ∈ R+ ,
(1) (2)
with constant coefficients a, b, h1 , h2 , h1 > 0, h2 > 0 and initial data 1
u(0, x) = u0 (x), ut (0, x) = u1 (x); u0 (x) ∈ H 1 (R), u1 (x) ∈ L2 (R), (−∆)− 2 u1 (x) ∈ L2 (R).
(3)
Here (−∆)−s u = F −1 (|ξ|−2s F(u)), for s > 0 and F(u), F −1 (u) are the Fourier and the inverse Fourier transform. By a solitary wave we mean a travelling with velocity c wave of the form ϕc (x − ct), vanishing at infinity. Let us recall, that a solitary wave ϕc (x − ct) is said to be orbitally stable if any solution u(t, x) of (1) – (3) with initial data close to the solitary wave for t = 0 stays close to ϕc (x − ct) for every t > 0. ∗ Corresponding
author Email addresses: [email protected] (N. Kolkovska), [email protected] (M. Dimova), [email protected] (N. Kutev) Preprint submitted to Mathematics and Computers in Simulation February 17, 2016
The orbital stability of the solitary waves to the generalized Boussinesq equation utt − u xx + h1 u xxxx + f (u) xx = 0, with a single nonlinearity
x ∈ R, t ∈ R+
f (u) = |u| p−1 u, p > 1
(4) (5)
2 and h1 = 1 is obtained by Bona and Sachs [2] for 1 < p < 5 and p−1 4 < c < 1. Later on, Liu [12] p−1 2 proves instability result for the same problem for c ≤ 4 and 1 < p < 5 or c2 < 1 and p ≥ 5. Strong instability to (4), (5), i.e. instability by means of blow up of the solutions, is obtained in p−1 . [13, 14] when 0 < c2 < 2(p+1) The stability of solitary waves to the generalized Boussinesq equation (4) with h1 = 1 and quadratic-cubic nonlinearity (2) is analytically studied in [5]. The authors prove that for some values of the constants a and b all solitary waves are stable (see Table 2). The double dispersion equation (1) with a single nonlinearity (5) is studied in [4]. The authors find conditions on c and the parameters a, b and p, for which the solitary waves are orbitally stable. Moreover, strong instability is proved for c2 < c20 , where the constant c0 is explicitly given. In the present paper we study the orbital stability or instability of the solitary waves to the double dispersion equation (1) with quadratic-cubic nonlinearity (2) and parameters, satisfying the conditions a > 0, b < 0, h1 > h2 . (6)
This choice is motivated by the investigations in [7, 8], where equation (4) with nonlinearity (2) has been proposed to be relevant to pulse propagation in biomembranes. Recently, the model (4), (2) is revised in [3] from the viewpoint of the solid mechanics. More precisely, a higher order term h2 uttxx with h2 - a small positive constant, h2 < h1 , is added to (4). With this term (4) transforms into the double dispersion equation (1). In both models [3, 7, 8], the constants in the equation are supposed to satisfy (6). For more details about the discussed models see [3, 7, 8] and the references therein. In this study we first derive explicitly all positive solitary waves to (1), (2). These waves are obtained in four regimes, depending on the relations between parameters a, b, h1 and h2 . The orbital stability or instability of solitary waves is studied by constructing of a function d(c), related to some conservation laws of (1) – (3) and applying the convexity or concavity criterion in [2, 6, 12]. By a stability or instability region we mean the set of wave velocities c, for which the solitary waves are orbitally stable or unstable. In Theorem 2 we obtain an analytical formula for d′′ (c). This formula allows one to check the orbital stability or instability of the solitary waves with velocity c for every choice of the constants a, b, h1 and h2 , satisfying (6). As a consequence, in Theorem 3 we give a complete description of the regions of orbital stability or instability of the solitary waves to generalized Boussinesq equation (4), (2). In this way we improve the results in [5] for all admissible values of the velocity c (see remark 4). For double dispersion equation (1), (2) the orbital stability or instability of the solitary waves is studied in neighbourhoods of the end points of the admissible values of the velocity c, see Theorem 5. In the special case of the generalized Boussinesq equation and double dispersion equation with a single quadratic nonlinearity our results reduce to the wellknown ones in the literature. The paper is organized in the following way. In Section 2 explicit formulas for the solitary waves to (1), (2) and (4), (2) are obtained. An analytical formula for d′′ (c), which determines 2
the orbital stability or instability of the solitary waves to (1), (2), is derived in Section 3. As an application of this formula, in Section 4 a complete investigation of the stability or instability of the solitary waves to (4), (2) is done, while Section 5 deals with an application to the double dispersion equation (1), (2). For simplicity, throughout the paper, for functions depending on x and t we use the short notation kuk = ku(·, t)kL2 (R) . By x+ we denote the first truncated power function: x+ = 0, if x ≤ 0, and x+ = x, if x > 0. 2. Solitary waves to double dispersion equation The solitary waves ϕc (x − ct) = ϕc (ζ) of the double dispersion equation (1), (2) with constant velocity c are solutions to the problem (h1 − h2 c2 )ϕ′′c (ζ) − (1 − c2 )ϕc (ζ) + aϕ2c (ζ) + bϕ3c (ζ) = 0, ϕc (ζ) → 0 for |ζ| → ∞.
(7)
When h2 = 0 the solutions to (7) are wellknown (see e.g. [11]) and are given in the following statement. Lemma 1. There exists a unique (up to translation of the coordinate system) positive solitary wave ϕc (ζ) to (4), (2) with constant velocity c −1 s r 1 − c2 9 ϕc (ζ) = 3(1 − c2 ) a + a2 + b(1 − c2 ) cosh ζ , ζ = x − ct, 2 h1
(8)
when one of the following assumptions is satisfied: (i) b > 0, (ii) b < 0,
a ∈ R, c2 ∈ [0, 1); a > 0, c2 ∈ 1 +
2a2 9b +
,1 .
The following lemma gives conditions on the parameters h1 , h2 , a and b for existence of positive solitary waves to (1), (2). Lemma 2. There exists a unique (up to translation of the coordinate system) positive solitary wave ϕc (ζ) of (1), (2) with constant velocity c −1 s r 2 1 − c 9 , ζ = x − ct, (9) 2 2 2 2 ζ ϕc (ζ) = 3|1 − c | a sgn(1 − c ) + a + b(1 − c ) cosh 2 h1 − h2 c2
when one of the following assumptions is fulfilled: h (i) b > 0, a ∈ R, c2 ∈ 0, min 1, hh12 ; 2 2 (ii) b > 0, a < 0, h1 < h2 1 + 2a c2 ∈ max 1, hh12 , 1 + 2a 9b , 9b ; h 2 2 1 c2 ∈ 1 + 2a (iii) b < 0, a > 0, h1 > h2 1 + 2a 9b + , 9b + , min 1, h2 ; (iv) b < 0, a ∈ R, c2 > max 1, hh21 . 3
Proof. We apply [1, Theorem 5], which gives necessary and sufficient conditions for existence of positive solutions to (7). According to [1, Remark 6.2.(ii)], the sign condition (h1 − h2 c2 )(1 − c2 ) > 0 necessarily holds. We look for positive solutions to (7) of the form r r h1 − h2 c2 h 1 − h 2 c2 = ψ(η), ζ = ϕc (ζ) = ϕc η η, 1 − c2 1 − c2
(10)
(11)
where ψ(η) satisfies the equation ψ′′ (η) − ψ(η) +
b a ψ2 (η) + ψ3 (η) = 0, ψ(η) → 0 for |η| → ∞. 2 1−c 1 − c2
(12)
From [10, Lemma 4.1] problem (12) has a unique (up to translation of the coordinate system) positive solution when either b > 0, a ∈ R (13) 1 − c2 or b 9b a a2 + (14) < 0, > 0, > 0. 2 2 2 1−c 1−c 2 1 − c2 1 − c2 This solution is given by the formula s −1 a 9b a2 + + ψ(η) = 3 cosh(η) 2 2 2 1−c 2 1−c 1 − c2 r −1 9b 2 2 2 2 1 − c cosh(η) . = 3|1 − c | a sgn(1 − c ) + a + 2
Using (11), we return back to function ϕc and obtain the form (9) of the unique positive solution to (7). Let us consider the restrictions on the parameters a, b, h1 , h2 and the velocity c, which guarantee the existence of solitary waves. Inequalities (10) and (13) are equivalent to
or
b > 0, 1 − c2 > 0, h1 − h2 c2 > 0, a ∈ R,
(15)
b < 0, 1 − c2 < 0, h1 − h2 c2 < 0, a ∈ R.
(16)
1 − c2 < 0, b > 0, a < 0, 2a2 + 9b(1 − c2 ) > 0, h1 − h2 c2 < 0
(17)
1 − c2 > 0, b < 0, a > 0, 2a2 + 9b(1 − c2 ) > 0, h1 − h2 c2 > 0.
(18)
Conditions (15) coincide with assumptions in Lemma 2(i), while conditions (16) - with assumptions in Lemma 2(iv). Analogously conditions (10) and (14) hold when
or
Conditions (17) lead to assumptions in Lemma 2(ii), while (18) is equivalent to assumptions in Lemma 2(iii). Lemma 2 is proved. 4
2
Remark 1. When condition (iii) of Lemma 2 holds and additionally 1 + 2a 9b < 0, then there exists a solitary wave ϕc with zero velocity c = 0, i.e. a ground state solution to (1), (2). The same observation is true when condition (ii) in Lemma 1 holds. Remark 2. For h1 = h2 = 1 the results in Lemma 2(i) and Lemma 2(iii) coincide with the results in [5, Theorem 1]. 3. Main results Throughout the rest of the paper we focus our investigation on the case ! ! 2a2 ,1 , b < 0, a > 0, h1 > h2 , c2 ∈ 1 + 9b +
(19)
i.e. we suppose that the conditions in Lemma 2(iii) are satisfied. We take into account, that problem (1), (2) with assumptions (19) is directly applicable to the model of pulse propagation in biomembranes, as it is mentioned in the Introduction. The other possible combinations of parameters, for which solitary waves to (1), (2) exist, i.e. the cases in Lemma 2(i), Lemma 2(ii), and Lemma 2(iv), will be treated in a forthcoming paper. In this section we formulate and prove the main results in the present paper. In order to study orbital stability or instability of the solitary waves ϕc (x − ct) we use the results from [2, 6, 12]. The proof is based on convexity or concavity of a function d(c) related to some conservation laws for equation (1). We rewrite problem (1), (2) to the following equivalent form ut = w x , wt = E − h2
∂2 ∂x2
!−1
E − h1
! ! ∂2 2 3 u − (au + bu ) x x , ∂x2
(20)
where E is the identity operator. The energy and momentum functionals of the system, given by ! Z u4 u3 1 dx, (21) u2 + h1 (u x )2 + w2 + h2 (w x )2 − 2a − b H(u, w) = 2 R 3 2 Z (uw + h2 u x w x ) dx, (22) M(u, w) = R
are conserved in time. We define also the functionals Ic (u) and Jc (u) as
Z au3 + bu4 dx, Ic (u) = h1 − h2 c2 ku x k2 + 1 − c2 kuk2 − R
! Z 1 au3 bu4 1 2 2 2 2 Jc (u) = dx. h1 − h2 c ku x k + 1 − c kuk − + 2 2 3 4 R
As in [2, 6, 12], we define the function d(c) (called ’the moment of instability’) as d(c) := H(ϕc , −cϕc ) + cM(ϕc , −cϕc ), with ϕc , H and M given by the expressions (9), (21) and (22) respectively. Now we formulate the following theorem: 5
Theorem 1. Suppose the parameters h1 , h2 , a and b are fixed and satisfy condition (19). Then d(c) = Jc (ϕc ),
(23)
where ϕc is the positive solitary wave to (1), (2) defined in (9). The function d(c) is given by the expression √ !1 1 4 h1 (1 − c2 ) 2 h2 2 2 d(c) = 1− c 9b2 h1 √ 2 p √ 2 2) 2a 2a + 9b(1 − c ) 2a + 3 −b(1 − c 2 log √ . (24) a + 3b(1 − c2 ) − p 1 1 12(−b) 2 (1 − c2 ) 2 2a − 3 −b(1 − c2 )
Proof. It is easy to check that the following identity
H(u, w) + cM(u, w) = 0.5kw + cuk2 + 0.5h2kw x + cu x k2 + Jc (u) holds. Since d(c) = H(ϕc , −cϕc ) + cM(ϕc , −cϕc ) = Jc (ϕc ),
then (23) is proved. We evaluate explicitly the function d(c). Since the solitary wave ϕc satisfies Ic (ϕc ) = 0, then Z Z 1 1 1 3 ϕ (x) dx + b ϕ4 (x) dx. (25) d(c) = Jc (ϕc ) = Jc (ϕc ) − Ic (ϕc ) = a 2 6 R c 4 R c q 1−c2 x we obtain from (25) the equality After the change of the variable s = exp h1 −h2 c2
Z ∞ s2 ds a d(c) =36(1 − c ) (h1 − h2 c ) 3 2 9 (a2 + b(1 − c2 )) 2 0 (s − 2νs + 1)3 2 Z ∞ 2 9b(1 − c ) s3 ds a + , where ν = − 1 . (26) 9 2 − 2νs + 1)4 2 2 2 (s (a + 2 b(1 − c )) 0 a2 + 92 b(1 − c2 ) 2 R ∞ sm−1 ds R∞ ds We evaluate integrals Pm = 0 (s2 −2νs+1) . m , m = 3, 4, by means of the integrals Qn = 0 (s2 −2νs+1)n Using the recurrence relation 2
5 2
2
1 2
2n − 3 ν + Qn−1 , n = 2, 3, 4, ... 2 2(n − 1)(1 − ν ) 2(n − 1)(1 − ν2 ) we get from (26) the expression d(c) as a function of Q1 : √ ! 12 !1 1 9 4 h1 (1 − c2 ) 2 h2 2 2 2 2 2 2 3b(1 − c ) + a − a a + b(1 − c ) Q1 . d(c) = 1− c 2 9b h1 2 Qn =
(27)
Under condition (19) we have ν2 =
√ 1 ν − ν2 − 1 a2 > 1 and Q = log . √ √ 1 a2 + 92 b(1 − c2 ) 2 ν2 − 1 ν + ν2 − 1
Finally, substituting Q1 in (27), we get the formula (24) for d(c). The proof of Theorem 1 is completed. 6
Let us consider the limiting case of (1), (2) with b = 0. In this case the double dispersion equation (1) transforms to problem with quadratic nonlinearity f (u) = au2 . As a consequence of Theorem 1 we have the following result. Corollary 1. Suppose the parameters of the double dispersion equation (1), (2) satisfy h1 > h2 , a > 0, b = 0 and c2 ∈ (0, 1). Then the function d(c) is reduced to the expression √ !1 6 h1 h2 2 2 2 25 (1 − c ) d(c) = 1 − . (28) c h1 5a2 Proof. In order to prove (28) we evaluate limb→0,b<0 d(c). We expand the logarithmic function in (24) in series around b = 0. We substitute the result into (24) and get formula (28). Let us note that (28) is identical with formula (4.1) from√[4] for p = 2 and a = 1. Indeed, for equation (1) with f (u) = au2 we have from [9] that d(0) = 65ah2 1 . In this way (28) transforms into formula (4.1) from [4]. According to the results in [2, 6, 12] the orbital stability or instability of the solitary waves to equation (1), (2) depends on the sign of the second derivative of the function d(c). In the following theorem we obtain an explicit formula of the second derivative d ′′ (c). Theorem 2. Suppose the parameters h1 , h2 , a and b are fixed and satisfy condition (19). Then the positive solitary wave ϕc of (1), (2) defined in (9) is orbitally stable if d′′ (c) > 0 and orbitally unstable if d ′′ (c) < 0, where p √ √ √ √ 2a + 3 −b 1 − c2 1 − c2 2a 4 h1 ′′ f (c) + √ g(c) log √ d (c) = p , 3 2 2 2 h2 2 2 2a + 9b − 9bc 12 −b 2a − 3 −b 1 − c 2 9b 1 − h1 c (29) ! h2 h2 h2 h2 2 h2 4 c a − 15a2b − 27b2 + 162b2 + 87a2 b + 378b2 h1 h1 h1 h1 h1 h22 h2 h2 4 2 2 2 2 h2 + −54a b 2 − 513b − 243b 2 c + 324b2 22 c6 , h1 h1 h1 h1
f (c) = − 18a2 b − 81b2 − 2
g(c) =18b + 2a2
h2 h2 h2 h2 + 9b − 81b c2 + 54b 22 c4 . h1 h1 h1 h1
Proof. Now we apply the result in [6] for orbital stability or instability of solitary waves to problem (20). The nonlinear term f (u) = au2 + bu3 satisfies all conditions of [12, Lemma 2.1] under assumptions (iii) of Lemma 2. Following the ideas in [2, 6, 12], one can prove that the stability or instability statements in [2, 6] holds also for the double dispersion equation (1), (2). Tedious calculations of derivatives give us from (24) formula (29). Theorem 2 is proved. Theorem 2 gives a direct method for studying orbital stability or instability of solitary waves ϕc of (1), (2). In order to check the stability or instability of the solitary waves one has only to substitute the values of the actual parameters a, b, h1 , h2 and c into formula (29). For the double dispersion equation (1) with a quadratic nonlinearity, as in Corollary 1, we evaluate the limit of d ′′ (c) for b → 0 with b < 0. 7
Corollary 2. Suppose the parameters of the double dispersion equation (1), (2) satisfy h1 > h2 , a > 0, b = 0 and c2 ∈ (0, 1). Then d ′′ (c) is given by the following formula d′′ (c) =
√ 1 6 h1 (1 − c2 ) 2 1 − 5a2 !2 h2 6 30 h1 c − 15
!− 3 h2 2 2 c h1 !2 ! h2 h2 4 h2 h2 2 − 5 + 51 c + 22 + 20 c − . h1 h1 h1 h1
(30)
4. Application to generalized Boussinesq equation
The investigation of the sign of d ′′ (c), i.e. the orbital stability or instability for the solitary waves to (1), (2) is a difficult problem because of numerous (exactly five) parameters. But, in the case of the generalized Boussinesq equation (4), (2) such examination can be done. In the following we set √ b k = 2 , ξ = 1 − c2 . a Then from Lemma 1(ii) and Remark 1 we have either ! 2 2 2 when k ≤ − , ξ ∈ 0, − 9k 9 or
2 < k < 0. 9 In the new variable ξ and for h2 = 0 formula (29) is rewritten as √ √ √ ! q 3(2 + 9k)ξ 2 2 + 3ξ −k 3a2 (−k) ′′ 2 =: G(k, ξ) 1 − ξ = 6ξ − − √ log √ √ √ d 2 + 9kξ2 4 h1 2 −k 2 − 3ξ −k ξ2 ∈ (0, 1] when −
(31)
In the following theorem we obtain explicit conditions on velocity c in terms of ab2 ∈ (−∞, 0) for which the solitary waves (8) are orbitally stable or unstable. We divide the admissible for ab2 interval (−∞, 0) into subintervals (−∞, k0 ), (k0 , − 29 ) and (− 92 , 0) because of the different behaviour of d′′ (c). The constant k0 is determined as the positive root of equation (37) and k0 ≈ −0.225397 < − 29 . Theorem 3. Suppose the parameters h1 , a and b are fixed and satisfy condition (19) with h2 = 0. If ϕc (x − ct) is the positive solitary wave to (4), (2), defined in (8), then the following statements hold for ϕc : 2
(i) If ab2 ∈ (−∞, k0 ), then all positive solitary waves ϕc with velocity c2 ∈ (1 + 2a 9b , 1) are orbitally stable. The value k0 is determined as the positive root of equation (37) and k0 ≈ −0.225397; 2
2 2 (ii) If ab2 ∈ (k0 , − 29 ), then all positive solitary waves ϕc with velocity c2 ∈ (1 + 2a 9b , c1 ) ∪ (c2 , 1) 2 2 2 are orbitally stable, while for c ∈ (c1 , c2 ) are orbitally unstable. Here the constants c1 , c2 are defined in (38);
8
(iii) If ab2 ∈ (− 92 , 0), then all positive solitary waves ϕc with velocity c2 ∈ (c23 , 1) are orbitally h stable, while for c2 ∈ 0, c23 are orbitally unstable. The constant c3 is defined in (41);
(iv) If ab2 = − 92 , then all positive solitary waves ϕc with velocity c2 ∈ (0, c24 ) are orbitally unstable, while for c2 ∈ (c24 , 1) are orbitally stable. The value of c4 is defined in (42) and c24 ∈ (0, 12 ).
Proof. According to Theorem 2, the sign of d′′ (c) determines the orbital stability or instability of solitary waves ϕc . Since the sign of d′′ (c) coincides with the sign of the defined in (31) function G(k, ξ), we study the behavior of the function G(k, ξ). Straightforward computations give us h i 27k ∂G (k, ξ) = 18kξ4 + (8 + 9k)ξ2 − 2 , 2 2 ∂ξ (2 + 9kξ ) h i 54kξ ∂2 G (k, ξ) = −81k2 ξ2 + 54k + 16 . 2 2 3 ∂ξ (2 + 9kξ )
(32)
From (32) it follows that
∂2 G 8 (k, ξ) > 0 for k < − and every ξ ≥ 0. ∂ξ2 27 Moreover, the equation
∂G ∂ξ (k, ξ)
(33)
= 0, considered as a polynomial of ξ, has real roots
p 1 (34) 8 + 9k ± (8 + 9k)2 + 144k 36k √ √ when either k ≤ − 89 2 + 3 or − 89 2 − 3 ≤ k < 0. 2 Case: k < − 92 , ξ2 ∈ 0, − 9k √ √ 8 8 , − 27 ≤ k ≤ − 98 2 − 3 and − 89 2 − 3 < This case is reduced to three subcases: k < − 27 k < − 29 . ξ±2 (k) = −
8 < k0 (35) 27 27 8 ∂G Since G(k, 0) = 0, ∂G ∂ξ (k, 0) = − 2 k > 0, from (33) we get for k < − 27 that ∂ξ (k, ξ) > 0, 8 G(k, ξ) > 0 for every ξ > 0 and hence d′′ (c) > 0. Thus for k < − 27 statement (i) of Theorem 3 follows from Theorem 2. Subcase: k < −
√ 8 8 ≤ k ≤ − 2 − 3 < k0 27 9 √ 40 Using the estimate 0 < 3 ≤ 9k + 16 ≤ 8 3, we obtain !2 2 ∂G 27 192 − (9k + 16) 8 + 9k 2 (k, ξ) = + ≥ 0 18 kξ + ∂ξ (2 + 9kξ2 )2 36 72 Subcase: −
(36)
2 ) and equality holds only for ξ2 = 8+9k for every ξ2 ∈ (0, − 9k 36k . Thus G(ξ, k) > 0 for every 2 2 ξ ∈ (0, − 9k ), when k is in the interval (36). Statement (i) is proved for k satisfying (36). 9
G(k0 , ξ)
G(k, ξ)
ξ1 (k) 0
•
•
•
•
ξ+ (k)
ξ− (k)
ξ2 (k) •
1
ξ •
0
(a) Figure 1: Function G(k, ξ), k =
•
ξ− (k0 )
ξ+ (k0 )
•
1
ξ
(b) b , a2
ξ=
√
√ 1 − c2 , for: (a) k ∈ − 89 (2 − 3), − 29 ; (b) k = k0 , k0 ≈ −0.225397.
√ 2 8 2− 3
i 54kξ± (k) h ∂2 G (k, ξ± (k)) = −81k2 ξ±2 (k) + 54k + 16 2 2 3 ∂ξ (2 + 9kξ± (k)) p p 27kξ± (k) (8 + 9k)2 + 144k (8 + 9k)2 + 144k ± 9k = 2 3 2(2 + 9kξ± (k))
and k < − 29 , it follows that ∂2 G (k, ξ− (k)) <0, ∂ξ2
p −1 432kξ+ (k) (8 + 9k)2 + 144k p ∂2 G (k, ξ+ (k)) = (8 + 9k)2 + 144k − 9k (2 + 9k) > 0. 2 2 3 ∂ξ (2 + 9kξ+ (k))
Hence ξ− (k) is a point of local maximum of G(k, ξ), while ξ+ (k) is a point of local minimum for G(k, ξ). q
2 2 From (31) we have G(k, 0) = 0, ∂G ∂ξ (k, 0) > 0 and from (2+9kξ )G(k, ξ) → −(2+9k) − k > 0 q as ξ → − 9k2 we conclude that ξ− (k) is a point of positive local maximum, while ξ+ (k) is a point 2 of local minimum for G(k, ξ) for ξ2 ∈ 0, − 9k . The value of the local minimum G(k, ξ+ (k)) can be a positive, negative or equal to zero number. Analysing the sign of G(k, ξ+ (k)) we conclude that there exists a constant k0 , which is a unique root of the equation
G(k, ξ+ (k)) = 0, where ξ+2 (k) = −
p 1 8 + 9k + (8 + 9k)2 + 144k , 36k
i.e. G(k0 , ξ+ (k0 )) = 0 (see Fig. 1(b)). Numerically we find out that k0 ≈ −0.225397. Hence, it follows that: 10
(37)
G(- 92 , ξ)
G(k, ξ)
•
0
ξ− (k)
1
•
•
ξ0 (k)
å
0
ξ
(a)
2 2
•
ξ0
1
•
ξ
(b)
Figure 2: Function G(k, ξ), k =
b , a2
ξ=
√
1 − c2 , for: (a) k ∈ − 92 , 0 ; (b) k = − 29 .
√ 2 • if − 98 (2 − 3) < k < k0 , then G(k, ξ) > 0 for every ξ2 ∈ (0, − 9k ) and the statement (i) is proved for this interval for k. Combining this result with the results in the previous subcases (35) and (36), we obtain the proof of statement (i) of Theorem 3. • if k0 < k < − 92 , then there exist points ξ1 (k), ξ2 (k), such that G(k, ξ1 (k)) = 0 and G(k, ξ2 (k)) = 0. 2 2 , G(k, ξ) > 0 for ξ ∈ (0, ξ1 (k)) ∪ (ξ2 (k), − 9k ) Moreover, ξ− (k) < ξ1 (k) < ξ+ (k) < ξ2 (k) < − 9k and G(k, ξ) < 0 for ξ ∈ (ξ1 (k), ξ2 (k)). (see Fig. 1(a).)
From the identity ξ2 = 1 − c2 and the sign of G(k, ξ), i.e. the sign of d′′ (c), statement (ii) is proved with constants c1 and c2 defined as c22 = 1 − ξ12 (k), Case: −
c21 = 1 − ξ22 (k).
(38)
2 < k < 0, ξ2 ∈ (0, 1] 9
In this case the function ∂G ∂ξ (k, ξ) has two real roots given in (34). Straightforward computations show that ξ+2 (k) > 1. Moreover, ξ−2 (k) = 1 −
p p −1 1 8 + 45k − (8 + 9k)2 + 144k = 1 − 6(2 + 9k) 8 + 45k + (8 + 9k)2 + 144k 36k
8 8 and for − 92 < k ≤ − 45 from the first equality and for − 45 < k < 0 from the second equality we 2 get ξ− (k) < 1. Since from (32) we have
h i ∂2 G 54kξ 54(−k)ξ 2 (k, ξ) = −81k2 ξ2 + 54k + 16 ≤ 4ξ + 12 − 16 2 2 3 2 3 ∂ξ (2 + 9kξ ) (2 + 9kξ ) 216(−k)ξ(ξ2 − 1) <0 = (2 + 9kξ2 )3 for every ξ ∈ (0, 1), it follows that G(k, ξ) is a strictly concave function w.r.t. ξ ∈ (0, 1). Thus ∂G ∂G 27 ∂ξ (k, ξ) has only one zero at ξ− (k) ∈ (0, 1) and from ∂ξ (k, 0) = − 2 k > 0 we get that ∂G ∂G (k, ξ− (k)) = 0; (k, ξ) > 0 for ξ ∈ (0, ξ− (k)); ∂ξ ∂ξ 11
∂G (k, ξ) < 0 for ξ ∈ (ξ− (k), 1]. ∂ξ
(39)
Tedious computations give us √
√ 2 + 3 −k G(k, 1) =3 − √ log √ √ <0 2 −k 2 − 3 −k 2
√
for k ∈ (− 29 , 0). Since G(k, 0) = 0 we get from (39) and G(k, 1) < 0 that there exists a unique point ξ0 (k), ξ02 (k) ∈ (ξ−2 (k), 1) such that G(k, ξ0 (k)) = 0; G(k, ξ) > 0 for ξ ∈ (0, ξ0 (k)); G(k, ξ) < 0 for ξ ∈ (ξ0 (k), 1]
(40)
(see Fig. 2(a)). The value of ξ0 (k) is defined from the equation √ √ √ 3(2 + 9k)ξ0 (k) 2 2 + 3ξ0 (k) −k log G(k, ξ0 (k)) = 6ξ0 (k) − − √ √ √ = 0. 2 + 9kξ02 (k) 2 −k 2 − 3ξ0 (k) −k For
c23 = 1 − ξ02 (k)
(41)
and (40) the proof of statement (iii) of Theorem 3 is completed. 2 Case: k = − , ξ2 ∈ (0, 1) 9 Now we investigate the special case k = − 92 . We have ! 2 3 1+ξ G − , ξ = 6ξ − log . 9 2 1−ξ 6ξ 2 3 ∂2 G 2 Since ∂G follows that G − 29 , ξ is a concave ∂ξ − 9 , ξ = 6 − 1−ξ2 and ∂ξ2 − 9 , ξ = − (1−ξ2 )2 < 0, it √ function, which has a positive maximum at the point ξ = 22 and limξ→1 G − 92 , ξ = −∞ (see √ Fig. 2(b)). Hence there exists a unique root ξ0 , ξ0 ∈ 22 , 1 , of the equation
1+ξ = 0, 1−ξ such that G − 92 , ξ > 0 for ξ ∈ (0, ξ0 ) and G − 92 , ξ < 0 for ξ ∈ (ξ0 , 1). Thus statement (iv) of Theorem 3 follows from Theorem 2 with constant c4 defined from 4ξ − log
c24 = 1 − ξ02 .
(42)
The proof of Theorem 3 is completed. Remark 3. If we substitute h2 = 0 in (30), we get the following expression for d′′ (c) √ 1 6 h1 lim d′′ (c) = (1 − c2 ) 2 20c2 − 5 2 b→0,b<0 5a
for the generalized Boussinesq equation (4) with quadratic nonlinearity f (u) = au2 . Since d′′ (c) < 0 for c2 < 41 and d′′ (c) > 0 for c2 > 14 , then, using Theorem 2, we establish the wellknown results for orbital stability or instability of solitary waves proved in [2, 12]. 12
Table 1: Regions of stability or instability of solitary waves to generalized Boussinesq equation (1) according to Theorem 3. The constants c1 , c2 , c3 , c4 and k0 are defined in the proof of Theorem 3 and k0 ≈ −0.225397.
Th. 3 b ∈ a2 c2 ∈
(i) (−∞, k0 ) 2
2
(ii) (k0 , − 92 )
(iv) − 92
(iii) (− 92 , 0)
2a 2 (1 + 2a (c21 , c22 ) (c22 , 1) (0, c24 ) (c24 , 1) [0, c23 ) (c23 , 1) 9b , 1) (1 + 9b , c1 ) stable stable unstable stable unstable stable unstable stable
Let us recall the result from [5] for orbital stability or instability of the solitary waves to problem (4), (2). The statement is suitably altered to fit the present context. Theorem 4 ([5]). Suppose b < 0, a > 0 are constants and h1 = 1.
(i) If ab2 ∈ (−∞, − 31 ], then all positive solitary waves of (4), (2) with velocity c2 ∈ 1 + are orbitally stable;
2a2 9b
,1
(ii) If ab2 ∈ (− 29 , 0), then the positive solitary waves of (4), (2) are orbitally stable for c2 ∈ (c2♯ , 1) and orbitally unstable for c2 ∈ (0, c2♭ ), where c♯ and c♭ are some constants satisfying the inequalities 0 < c♭ < c♯ < 1. Remark 4. Let us compare the results in Theorem 3 with the corresponding ones in Theorem 4. For convenience, we illustrate the results in Theorem 3 and Theorem 4 by means of Table 1 and Table 2, correspondingly. • the case
b a2
∈ (− 31 , − 29 ] is studied in Theorem 3 for the first time.
• in case ab2 ∈ (− 92 , 0) the results in Theorem 3 improve the result in [5]. More precisely, in [5] the orbital stability or instability is investigated only in a small neighbourhood of c = 0 and c = 1, while in Theorem 3 the result is extended to the whole interval [0, 1) for c. Remark 5. The results for stability or instability of solitary waves are very sensitive with respect to the ratio ab2 for ab2 ∈ (− 31 , 0). For example when ab2 ∈ (k0 , − 92 ], i.e. an interval with length less than 0.0032, the stability or instability of solitary waves is quite different in comparison with the corresponding one in the neighbouring intervals (− 31 , k0 ) and (− 92 , 0). 5. Application to double dispersion equation In this section we apply Theorem 2 to double dispersion equation (1), (2). The results are not so comprehensive as in Theorem 3. They outline the influence of the ratio of the dispersive coefficients hh12 for orbital stability or instability of the solitary waves in neighbourhoods of the end points of the admissible values of c. Theorem 5. Suppose the parameters h1 , h2 , a and b are fixed and satisfy condition (19). If ϕc (x − ct) is the positive solitary wave of (1), (2), defined in (9), then the following assertions hold for ϕc : 13
Table 2: Regions of stability or instability of solitary waves to generalized Boussinesq equation (1) according to Theorem 4 in [5]. The constants c♭ , c♯ are defined in the proof of Theorem 4 and k0 is defined in the proof of Theorem 3, k0 ≈ −0.225397.
Th. 4([5]) b ∈ a2 c2 ∈
(i) (−∞, − 31 ]
(1 +
2a2 9b , 1)
stable
(− 31 , k0 )
(k0 , − 29 ]
2
2
(1 + 2a 9b , 1) not considered
(1 + 2a 9b , 1) not considered
(ii) (− 92 , 0) (0, c2♭ ) unstable
[c2♭ , c2♯ ] not considered
(c2♭ , 1) stable
(i) in a neighbourhood of the right-end point c = 1:
2 , 1 , such that for for every ab2 ∈ (−∞, 0) there exists a constant C1 , C12 ∈ 1 + 2a 9b + c2 ∈ C12 , 1 all positive solitary waves ϕc with velocity c are orbitally stable;
(ii) in a neighbourhood of the left-end point c = 0 or c = 1 + (ii)a if
b a2
2a2 9b
:
∈ (−∞, − 92 ), then there exists a constant C2 , C22 ∈ 1 + 2
2a2 9b , 1
, such that for
2 c2 ∈ (1 + 2a 9b , C2 ) all positive solitary waves ϕc with velocity c are orbitally stable; (ii)b if ab2 ∈ − 29 , 0 , then there exists a constant C3 ab2 , hh12 , C32 ∈ (0, 1), such that all positive solitary waves ϕc with velocity c are orbitally unstable for c2 ∈ (0, C32 );
(ii)c if ab2 = − 29 , then there exists a constant C4 , such that all positive solitary waves ϕc with velocity c2 ∈ (0, C42 ) are orbitally unstable.
The constants Ci , i = 1, 2, 3, 4, depend on
b a2
and
h2 h1 .
Proof. We use the short notations √ h2 , ξ = 1 − c2 . h1 Note that µ ∈ (0, 1) and for k ≤ − 92 we get ξ2 ∈ 0, − 9k2 , while for k ∈ (− 29 , 0) we have ξ2 ∈ (0, 1]. Using the new notations, we rewrite formula (29) as √ √ ! 3 5 q 2 + 3ξ −k 9a2 (−k) 2 (1 − µc2 ) 2 ′′ 2 d 1 − ξ = G(k, ξ, µ) = G1 (k, ξ, µ) log √ √ √ + G2 (k, ξ, µ), 4 h1 2 − 3ξ −k (43) k=
with
b , a2
µ=
√
2 18k + 2µ(1 − 36k) + 54kµ2 + 27kµ(3 − 4µ)ξ2 + 54kµ2 ξ4 , 12 2 + 9kξ2 = µ2 81k2 − 54k + (108k − 486k2 )ξ2 + (729k2 − 54k)ξ4 − 324k2 ξ6 G2 (k, ξ, µ) √ −kξ + µ −162k2 + 72k − 2 + (648k2 − 87k)ξ2 − 513k2 ξ4 + 81k2 − 18k − 162k2 ξ2 ,
G1 (k, ξ, µ) =
14
Proof of (i) Simple computations lead to 5
G(k, 0, µ) = 0,
∂G 81(−k) 2 (k, 0, µ) = (1 − µ)2 > 0 ∂ξ 2
and from (43), (44) we get that d′′ (1) > 0. Hence, there exists a constant C1 , C12 ∈ 1 + such that d′′ (c) > 0 for c2 ∈ C12 , 1 This proves assertion (i) of Theorem 5. Proof of (ii)a If k = Since
b a2
√
√ 2 + 3ξ −k =0 lim √ √ 2 (2 + 9kξ ) log √ 2 − 3ξ −k ξ→ − 9k
2a2 9b +
< − 92 , we calculate the limit of (2 + 9kξ2 )G(k, ξ, µ) as ξ → 2
it follows that
(44) ,1 ,
q 2 . − 9k
2 lim √ 2 (2 + 9kξ )G(k, ξ, µ) =
ξ→
− 9k
√ ! 2 2 8 2 2 2 µ (81k + 54k + 12 + ) + µ(−162k − 72k − 8) + 81k + 18k) 3 9k √ ! µ2 2 (9k + 2) 9(1 − µ)2 k − 4µ(1 − µ) + >0 = 3 9k
for k = ab2 < − 29 and µ ∈ (0, 1). 2 Hence, there exists a constant ξ2 , 0 < ξ22 < − 9k2 such that G(k, ξ, µ) > 0 for ξ2 ∈ (ξ22 , − 9k ). If 2 2 ′′ 2 2 C2 = 1 − ξ2 , then we get d (c) > 0 for c ∈ (0, C2 ) and statement (ii)a in Theorem 5 is proved. Proof of (ii)b In order to prove (ii)b we calculate √ √ √ 5 √ 9(−k) 2 ′′ 2 2 + 3 −k (18k + 2µ + 9kµ) log √ √ d (0) = − −k(9k + 3kµ + µ) + √ = G(k, 1, µ). 12 4 h1 2 − 3 −k We evaluate also the derivatives √ ∂G 1 (k, 1, µ) = √ (9k + 3kµ + µ) − −k(9 + 3µ) ∂k 2 −k √ √ √ 2 2 + 3 −k 18k + 2µ + 9kµ + (6 + 3µ) log √ √ − √ 4 2 − 3 −k 2 −k(2 + 9k) √ √ √ √ √ 3 2 2 + 3 −k 9 9 −k (2 + µ) log √ , = √ − (3 + µ) −k + 4 2 + 9k 2 − 3 −k 2
√ ∂2 G 9(3 + µ) 9 81 −k 9(2 + µ) + − − (k, 1, µ) = − √ √ √ ∂k2 −k(2 + 9k) 4 −k 2 −k(2 + 9k) (2 + 9k)2 9 = √ −4(2 + µ)(2 + 9k) + (3 + µ)(2 + 9k)2 − 2(2 + 9k) + 36k 4 −k(2 + 9k)2 18 ((2 + 9k)(27k + 9kµ − 2µ) + 36k) < 0. = √ 4 −k(2 + 9k)2 15
b a2
d′′ (c)
= −0.23
d′′ (c)
h2 h1
h2 h1 =0.9
= 0.1
h2 h1 =0.9 h2 h1
= 0.1
0
1
|
0
0.1839 ≈
q
1+
2a2 9b
b a2
1 c
(a) Figure 3: Function d′′ (c) for different values of
c
= −0.21
(b) h2 h1
and: (a) k =
b a2
= −0.23 < − 92 ; (b) k =
b a2
= −0.21 > − 92 .
Hence G(k, 1, µ) is a strictly concave function w.r.t. k for k ∈ (− 92 , 0). Moreover, there exists a 2 constant ξ3 such that ∂∂kG2 (k, ξ, µ) < 0 for ξ2 ∈ (ξ32 , 1] and k ∈ (− 92 , 0). From ∂G ∂k (0, 1, µ) = 0 it 2 2 (k, ξ, µ) > 0 for ξ ∈ (ξ , 1]. follows that ∂G 3 ∂k Thus G(k, ξ, µ) is a strictly increasing function w.r.t. k for ξ2 ∈ (ξ32 , 1]. From G(0, 1, µ) = 0 we get G(k, ξ, µ) < 0 for ξ2 ∈ (ξ32 , 1], i.e. we have d ′′ (c) < 0 for k ∈ (− 92 , 0), c2 ∈ (0, C32 ) with C3 defined as C32 = 1 − ξ32 . In this way we proved statement (ii)b in Theorem 5. Proof of (ii)c Let k = − 92 . In order to evaluate limξ→1 G(− 92 , ξ, µ) we calculate functions G1 and G2 , defined in (43), for k = − 29 , i.e. ! √ ! 2ξ 2 2 2 2 2 2 µ 16(1 − ξ ) + µ(38ξ − 39) + 8 , G2 − , ξ, µ = 9 6 3 √ ! 2 2 (45) 1 + 3µ2 − 6µ2 ξ2 + 3µ2 ξ4 . G1 − , ξ, µ = − 9 3 √ √ (12−µ) 2 2 2 . Hence lim G − , ξ, µ = − We get from (45) that limξ→1 G2 − 29 , ξ, µ = (12−µ) ξ→1 9 9 √ 9 1+ξ 2 2 From the continuity of G − 9 , ξ, µ w.r.t ξ, there exists ξ4 3 limξ→1 log 1−ξ = −∞ for µ ∈ (0, 1). 2 2 such that for ξ ∈ (ξ4 , 1] we have G − 92 , ξ, µ < 0. Applying Theorem 2, we get statement (ii)c of Theorem 5 for c2 ∈ (0, C42 ) and C4 defined from C42 = 1 − ξ42 . Theorem 5 is completely proved. Let us consider once again the limiting case b = 0, i.e. the double dispersion equation (1) with quadratic nonlinearity f (u) = au2 . Lemma 3. Suppose the parameters of the double dispersion equation (1), (2) satisfy h1 > h2 , a > 0, b = 0 and c2 ∈ (0, 1). Then there exists a constant c∗ , c2∗ ∈ (0, 1) such that all positive solitary waves ϕc to (1), (2) with velocity c2 ∈ (0, c2∗ ) are orbitally unstable, while with velocity c2 ∈ (c2∗ , 1) are orbitally stable. Proof. According to (30), the sign of d′′ (c) in the interval c2 ∈ (0, 1) depends only on the sign of the polynomial F(c2 ) !2 !2 ! h2 h2 h h2 h 2 2 2 6 F(c ) = 30 − 5. c − 15 + 51 c4 + 22 + 20 c2 − h1 h1 h1 h1 h1 16
d′′ (c)
- region of stability - region of instability
|
0
Figure 4: Function d′′ (c) and the regions of stability or instability of solitary waves for q 2 1 + 2a 9b ≈ 0.1007.
Let η = c2 and µ =
h2 h1 .
1
0.1007
h2 h1
c
= 0.9, k =
b a2
= −0.2245 and
Using these notations, we rewrite F as a new function F1 ,
F1 (η) := F(c2 ) = 30µ2 η3 − 15µ2 + 51µ η2 + (22µ + 20) η − µ − 5,
which is a third order polynomial of η. From the Sturm’s theorem for the number of the real roots of the equation F1 (η) = 0, (46) it follows that (46) has only one positive root in the interval η ∈ (0, 1). Indeed, the number of variations W(0) of the signs of the sequence F1 (0) = −µ − 5 < 0, F1′ (0) = 22µ + 20 > 0, F1′′ (0) = −30µ2 − 102µ < 0, F1′′′ (0) = 180µ2 > 0 is 3, i.e. W(0) = 3, for every µ ∈ (0, 1). Correspondingly, W(1) = 2 for the sequence F1 (1) = 15(µ − 1)2 > 0, F1′ (1) = 20(1 − µ)(1 − 3µ), F1′′ (1) = 6µ(25µ − 17), F1′′′ (1) = 180µ2 > 0, 17 17 calculated for each admissible value of µ: µ ∈ (0, 13 ), µ = 31 , µ ∈ ( 13 , 17 25 ), µ = 25 , µ ∈ ( 25 , 1) and neglecting the zero terms. Thus equation (46) has one real root η∗ in the interval (0, 1) because W(0) − W(1) = 1. Since F1 (0) is negative, we have for η < η∗ that F1 (η) < 0, while for η > η∗ we have F1 (η) > 0. The rest of the proof follows from Theorem 2 and Lemma 3 is proved for c2∗ = η∗ .
Note that the result in Lemma 3 is announced and numerically demonstrated in [4]. The analytical investigation of the regions of stability or instability of the solitary waves to (1), (2) is a quite complicated problem as it is clear from the proof of Theorem 5. However, for every particular choice of the parameters of (1), (2) the application of Theorem 2 simplifies this problem. Thus the regions of stability or instability of the solitary waves are obtained for every admissible choice of c. On Fig. 3 for two special choices of the quantities ab2 and hh12 the graphs of d′′ (c) are given. Figure 3(a) illustrates the statement of Theorem 5(i) and Theorem 5(ii)a , while Figure 3(b) illustrates Theorem 5(i) and Theorem 5(ii)b . 17
On Fig. 4 the graph of d′′ (c) for parameters hh12 = 0.9 and k = ab2 = −0.2245 is presented. One can observe that the graph of d′′ (c) is completely different from the graphs of d′′ (c) for values of k close to −0.2245 and fixed ratio hh21 = 0.9: • for k = −0.23 all solitary waves are stable, see Fig. 3(a); • for k = −0.21 the admissible region (0, 1) of the velocity c2 is divided into two subintervals. The first one is a region of instability and the second one - a region of stability, see Fig. 3(b); • for k = −0.2245 we have two regions of stability, divided by a region of instability, see Fig. 4. 6. Conclusions In this paper the orbital stability or instability of solitary waves to double dispersion equation (1) and generalized Boussinesq equation (4) with quadratic-cubic nonlinearity f (u) = au2 + bu3 , a > 0, b < 0 is studied. All positive solitary waves to the double dispersion equation are explicitly found. An analytical formula for d ′′ (c) is derived for double dispersion equation (1), (2). This formula allows one to obtain the regions of stability or instability for every particular choice of the parameters of problem (see Theorem 2). In the special case of generalized Boussinesq equation (4) the orbital stability or instability of the solitary waves is completely investigated for all admissible velocities c (see Theorem 3). In particular, the case ab2 ∈ (− 31 , − 92 ) is investigated for the first time in this study. For double dispersion equation (1), (2) the orbital stability or instability of the solitary waves is studied in neighbourhoods of the end points of the admissible values of the velocity c (see Theorem 5). The statements depend only on the ratios hh12 and ab2 . For h1 > h2 , the results do not essentially affected by the values of h2 . In the limiting case h2 = 0 and b = 0 the statement in Theorem 3 reduces to results in [2, 12]. When h2 , 0 and b = 0, Theorem 5 coincides with the result in [4], see Corollary 3. Acknowledgement This research is partially supported by the Bulgarian Science Fund under grant DFNI I-02/9. References [1] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of solitary waves, Arch. Rat. Mech. Anal., 82 (1983) 313–346. [2] J. Bona, R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys., 118 (1988) 15–19. [3] J. Engelbrecht, K. Tamm, T. Peets, On mathematical modelling of solitary pulses in cylindrical biomembranes, Biomech. Model. Mechanobiol., 14 (2015) 159–167. [4] H.A. Erbay, S. Erbay, A. Erkip, Instability and stability properties of traveling waves for the double dispersion equation, Nonlinear Analysis, 133 (2016) 1–14 [5] H. Freist¨uhler, J. H¨owing, An analytical proof for the stability of Heimburg-Jackson pulses, ArXiv:1303.5941v1 (2013) [6] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, Journal of Functional Analysis, 74 (1987) 160–197. [7] T. Heimburg, A.D. Jackson, On soliton propagation in biomembranes and nerves, Proc. Natl. Acad. Sci USA, 102 (2005), 9790–9795.
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[8] T. Heimburg, A.D. Jackson, On the action potential as a propagating density pulse and the role of anesthetics, 57 (2007) Biophys. Rev. Lett., 2 (2007) 57–78. [9] N. Kutev, N. Kolkovska, M. Dimova, Theoretical and numerical aspects for global existence and blow up for the solutions to Boussinesq paradigm equation, AIP Conf. Proc. 1404 (2011) 68–76. [10] N. Kutev, N. Kolkovska, M. Dimova, Global existence to generalized Boussinesq equation with combined powertype nonlinearities, J. Math. Anal. Appl. 410 (2014) 427-444. [11] B. Lautrup, R. Appali, A. Jackson, T. Heimburg, The stability of solitons in biomembranes and nerves, Eur. Phys. J. E, 34 (2011) 57–65 [12] Y. Liu, Instability of solitary waves for generalized Boussinesq equation, J Dynamics Differ. Equat., 5 (1993) 537–558. [13] Y. Liu, Instability and blow-up of the solutions to a generalized Boussinesq equation, SIAM J Math. Anal., 26 (1995) 1527–1546. [14] Y. Liu, M. Ohta, G. Todorova, Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations, Annales de l’Inst. Henri Poincare (c) Non-linear Analysis, 24 (2007) 539 –549.
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