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Numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems
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Numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems J. Álvarez a,b , A. Durán c,b,∗ a
Department of Applied Mathematics, University of Valladolid, Paseo del Cauce 59, 47011, Valladolid, Spain
b
IMUVA, Institute of Mathematics of University of Valladolid, Spain
c
Department of Applied Mathematics, University of Valladolid, Paseo de Belen 15, 47011, Valladolid, Spain
highlights • We propose a method to compute periodic traveling waves. • The method can be applied to some nonlinear dispersive systems. • It is checked in fractional KdV equations and Boussinesq systems.
article
abstract
info
Article history: Received 5 November 2015 Received in revised form 26 July 2016
In this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced. The method is based on a suitable modification of a fixed point algorithm of Petviashvili type and solves several drawbacks of some previous algorithms presented in the literature. By way of illustration, the method is applied to generate numerically periodic traveling waves of two problems of interest: the fractional KdV type equations and the extended Boussinesq system. © 2016 Elsevier B.V. All rights reserved.
MSC: 65H10 65M99 35C99 35C07 76B25 Keywords: Nonlinear dispersive equations Periodic traveling waves Iterative methods for nonlinear systems Petviashvili type methods Fractional KdV equation e-Boussinesq system
1. Introduction Many mathematical models for nonlinear dispersive wave propagation are governed by evolution systems of the form ut + Lux + f (u)x − Nut = 0,
x ∈ R, t ≥ 0.
(1)
In (1) u = u(x, t ) = (u1 (x, t ), . . . , um (x, t )) is a real-valued vector function (m ≥ 1) of x ∈ R, t ≥ 0, L and N are m × m matrices with entries given by Fourier multiplier operators with symbols lij (ξ ), nij (ξ ), i, j = 1, . . . , m, ξ ∈ R T
∗
Corresponding author at: Department of Applied Mathematics, University of Valladolid, Paseo de Belen 15, 47011, Valladolid, Spain. E-mail addresses: [email protected] (J. Álvarez), [email protected] (A. Durán).
http://dx.doi.org/10.1016/j.cam.2016.08.037 0377-0427/© 2016 Elsevier B.V. All rights reserved.
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respectively and f = (f1 , . . . , fm )T : Rm → Rm is smooth (see e. g. [1] and references therein). Much of the relevance of (1) is due to the possible existence of traveling wave solutions, [2]. They are characterized as waves of permanent form uj (x, t ) = φj (x − cs t ), j = 1, . . . , m, traveling with fixed speed cs ∈ R. In the study of existence and dynamics of such solutions, numerical computation plays a relevant role since typically the profiles φj are not known to exist or when the problem of existence is overcome, there are no exact formulas. This paper is concerned with the numerical generation of traveling wave solutions of (1) of periodic type, that is where the profiles φj are periodic of some period 2l, l > 0. The vector φ = (φ1 , . . . , φm ) must therefore satisfy the system
(−cs (I − N ) + L)φ + f (φ) = A,
(2)
where A = (A1 , . . . , Am ) is (nonzero in general) constant. (Note that, contrary to (1), system (2) is stationary since it only depends on the variable x − cs t.) The existence and dynamics of these solutions are relevant in their own and because of the relation with solitary wave solutions in the long-wavelength limit, [3]. Results on existence and stability for particular cases of (1) can be seen in the literature, see e. g. [4] and references therein. It is also worth mentioning some additional recent results. Thus, by way of modest illustration, Chen and Bona, [5], establish existence of periodic traveling wave solutions of (1) for the case m = 1, N = 0, under some hypotheses on f and L as well as stability under perturbations of the same period. (This study generalizes previous works on this matter, [6].) Also, Chen et al., [7], analyze the existence of periodic traveling waves of systems of Boussinesq type. On the other hand, Mikyoung Hur and Johnson, [8], study periodic traveling waves for KdV equations with fractional dispersion, analysis that may be generalized to models of the form (1) with m = 1, N = 0. Finally, Pipicano and Muñoz, [9], discuss the existence of these waves for a regularized Benjamin–Ono system. As far as the numerical generation is concerned, the literature emphasizes the use of pseudospectral Fourier approximation to (2) and different resolution of the resulting discrete equations, from the application of a Fourier–Galerkin algorithm as explicit analytical approximation, [10], to the use of iterative methods. Concerning this last approach, the application of classical methods like the fixed point algorithm is in general not convergent, [11], while the Newton method (used e. g. in [9]) suffers from important drawbacks, [12]. In this sense, in [13] a numerical scheme for the computation of periodic traveling waves of (1) with m = 1, N = 0 was proposed. This was based on a Fourier pseudospectral discretization in space of (1) along with time integration with conservation properties of invariant quantities of the periodic boundary-value problem associated to (1). The periodic traveling wave profile was approximated by using the iterative Petviashvili method, [14,11]. The resulting scheme ensured an accurate computation of the profile and speed of the waves. The present work is a continuation of [13] and attempts to complete it in several points. The first one is the extension of the study to systems of the form (1). The second one concerns the feature that the generation of the profile may involve nonzero constants A in (2) which make the Petviashvili method unable to be used. A theoretical strategy of reducing (2) to the case A = 0, considered in e. g. [15,5], forces to make a change in the nonlinearity which also prevents the application of this iterative scheme. Since homogeneity of degree greater than one in magnitude is required in [11] and the presence of constants is not considered in [16], the present paper complements in this sense the previous when applied to a new problem of numerical generation of periodic traveling waves. Indeed, the case of zero constant reduces to consider the method proposed in [16] and the case of zero constant and homogeneous (of degree greater than one in magnitude) nonlinear term deals with the application of the Petviashvili method to compute periodic traveling waves in the nonlinear dispersive wave equations under study. In order to overcome these two last drawbacks, the present paper introduces a two-fold strategy. This is based on dividing the solution into a constant plus a second profile. The constant is a solution of a polynomial system, which is iteratively solved, while the second profile can be approximated by a fixed point algorithm of Petviashvili type complemented by acceleration techniques to improve the performance, [17]. The structure of the paper is as follows. In Section 2 the proposed algorithm is described under some hypotheses on (1). The performance of the method is tested in Section 3 by considering the computation of periodic traveling waves in two models. They are the generalized fractional KdV equation, [1,18], and the e-Boussinesq system, [19]. In the first case, there are theoretical results of existence of waves but no exact formulas for the profiles. In the second case and to our knowledge, the problem of existence has not been studied yet. For this reason, the existence of such waves is implicitly assumed throughout the paper although, since the theoretical question is out of our goal, no specific hypotheses will be posed (see the above comments and mentioned references in this Introduction about this problem). Finally, Section 4 contains some concluding remarks. 2. Preliminaries and numerical method The numerical technique below is applied to compute periodic traveling wave solutions of (1) under two assumptions: (H1) For each (i, j), 1 ≤ i, j ≤ m the Fourier symbols lij and nij of the (i, j)th entries of the matrix operators L and N are real, even and continuous. (H2) For each j = 1, . . . , m, the nonlinear term fj is a polynomial in Rm with fj (0) = ∂∂x fj (0) = 0, k = 1, . . . , m and degree k pj − 1 with pj ≥ 3. As will be seen below, conditions (H1), (H2) are required for the derivation of the proposed numerical resolution of (2): while (H1) implies that any constant vector is in the kernel of the operators L and N, the use of (H2) allows to transform (2) into a
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new system where the nonlinear term only contains homogeneities of degree at least two. On the other hand, many of the wave propagation models governed by systems (1) presented in the literature involve linear and nonlinear terms satisfying (H1) and (H2) respectively. By way of illustration, scalar cases like the KdV, BBM, Benjamin–Ono and Benjamin equations along with their generalized versions (see e. g. [1,4] and references therein) or systems like the four-parameter Boussinesq equations, [20,21], belong to this family. On the contrary, important models like Serre equations, [22,23], are out of it. The method proposed here for the numerical resolution of (2) consists of the following steps: (S1) Finding (approximate or exact) real constant solutions φ = C = (C1 , . . . , Cm ) of (2) that is
(−cs (I − N ) + L)C + f (C ) = A.
(3)
(S2) The change of variables ϕ = φ − C , when φ is a solution of (2) and (3) lead to a system for ϕ of the form:
L(C )ϕ − N (ϕ) = 0,
(4)
with
L(C ) = cs − (L + cs N ) − f ′ (C ), N (ϕ) = (N1 (ϕ), . . . , Nm (ϕ))T Nk (ϕ) =
p−1 m 1 Djxi ,...,xi fk (C )ϕxi · · · ϕxi j 1 j 1 j ! i ,...,i =1 j =2 j
1
where p = max{p1 , . . . , pm } and Djxi ,...,xi fk (C ) = j 1
∂ j fk (C ), ∂ xi1 · · · ∂ xij
k = 1, . . . , m
(S3) System (4) is iteratively solved with an extended version of the Petviashvili method, [16]. The nonlinear term N is rewritten in the form
N (ϕ)
=
p−1
N (j) (ϕ),
(j)
N (j) (ϕ) = (N1 (ϕ), . . . , Nm(j) (ϕ))T
j =2
(j)
Nk (ϕ) =
m
1
j! i ,...,i =1 j 1
Djxi ,...,xi fk (C )ϕxi · · · ϕxi , 1 j j 1
k = 1, . . . , m.
Now each vector N (j) (ϕ) is homogeneous of degree j = 2, . . . , p − 1. Thus, from ϕ [0] ̸= 0 and for ν = 0, 1, . . . the following iteration is considered: p−1
L(C )ϕ [ν+1] =
sj (ϕ [ν] )N (j) (ϕ [ν] ),
ν = 0, 1, . . . ,
(5)
j =2
where sj (u) = s(u)αj ,
αj =
j j−1
,
s(u) =
⟨L(C )u, u⟩ , ⟨N (u), u⟩
(6)
j = 2, . . . , p − 1.
The so-called stabilizing factors sj in (6) are obtained by using the same quotient s(u) (where ⟨·, ·⟩ stands for the usual Euclidean inner product) but changing the exponent αj . (Note in particular that in the case of convergence, s(ϕ [ν] ) → 1.) Formulas (5) and (6) correspond to an extension of the classical Petviashvili scheme, [14], when the nonlinearity consists of several homogeneous terms with different degree of homogeneity. The procedure considered here takes advantage of the behavior of the Petviashvili method and its generalization for nonlinear systems with homogeneous nonlinearities of degree greater than one in magnitude. For such systems, the degree of homogeneity is an eigenvalue of the iteration matrix at the fixed point for the classical fixed point algorithm, which in general enables it to be convergent. The Petviashvili iteration acts like a filter of this harmful eigenvalue, in such a way that its corresponding iteration matrix at the fixed point transforms the degree of homogeneity to an eigenvalue of magnitude below one and preserves the rest of the spectrum of the iteration matrix of the classical fixed point algorithm, translating then the question of the convergence to this remaining part of the spectrum, [11]. This behavior was also experimentally observed in the extended version proposed in [16] for nonlinearities consisting of superposition of homogeneous terms of degree above one in magnitude. The polynomial system (3) can be iteratively solved by any of the algorithms developed in the literature to this end: for instance, Newton-type iterations, [24,25] (since in this case the drawbacks mentioned in [12] do not apply), iterative
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methods, [26,27], homotopy continuation, [28,29] or symbolic computational procedures like Gröbner bases, [30], resultant elimination methods, [31] or Wu’s method, [32]. The operator (2) taking part of L(C ) in (4) suggests to formulate (5), (6) in Fourier space for the Fourier coefficients [ν] , n ∈ Z of the iterations ϕ [ν] , ν = 0, 1, . . . Then for fixed c and l > 0 the problem (2) on (−l, l) with periodic boundary ϕ s n conditions is considered and (5), (6) take the form
cs I − f ′ (C ) + ( L + cs N)
nπ l
[ν+1] = ϕ n
cs I − f ′ (C ) + ( L + cs N)
n∈Z
sj (ϕ [ν] ) N (j) (ϕ [ν] ) n ,
n ∈ Z, ν = 0, 1, . . . ,
(7)
j=2
sj (u) =
p−1
nπ l
un , un
αj
un N (j) (u) n
,
j = 2, . . . , p − 1,
(8)
n∈Z
N )( nlπ ) denotes the Fourier symbol matrix with entries (lij ( nlπ ) + cs nij ( nlπ )), i, j = 1, . . . , m, αj is given by (6) where ( L + cs and un = (( u1 )n , . . . , ( um )n )T , n ∈ Z. In the experiments below, system (4) is discretized on an interval with a Fourier pseudospectral method as in [13], see [10]. This allows to implement the iteration by using the discrete Fourier version of (7), (8) and FFT techniques. Furthermore, in some of the experiments, the resulting scheme has been implemented with vector extrapolation methods to accelerate the convergence. The literature on these techniques is extensive, see e. g. [33,34] and references therein. (The use of vector extrapolation methods in traveling wave simulations is analyzed in [17].) 3. Numerical examples This section is devoted to illustrate the behavior of the algorithm to generate numerically periodic traveling waves. This will be shown by considering two models of the form (1): a generalized version of the fractional KdV equation and the extended Boussinesq system. Besides these illustrative purposes, there are additional reasons to justify the choice of these equations. In the first case, periodic traveling wave solutions are known to exist (under certain hypotheses) but without exact formulas for the profiles. Then the algorithm may be useful to compute accurate approximations, in order to be used, for example, in further analyses, by numerical means, of the dynamics. In the case of the second system, to our knowledge, the problem of existence of periodic traveling wave solutions has not been studied yet and the proposed algorithm may additionally serve as a preliminary numerical evidence of such waves in the model. 3.1. Generalized fractional KdV equation The first example treated here concerns the generation of periodic traveling waves of KdV models with fractional dispersion, of the form ut − Λµ ux + (f (u))x = 0,
(9)
where
v(ξ ) = |ξ | Λ v(ξ ),
0 < µ ≤ 2,
f (u) =
up−1 p−1
,
p ≥ 3.
Eq. (9) is of the form (1) with m = 1, N = 0, L = Λµ . Existence and stability of periodic traveling wave solutions are analyzed in [8,18] by using variational arguments. Specifically, existence is obtained for 0 < p < pmax where pmax = pmax (µ) :=
2 1−µ
+∞
if µ < 1
(10)
if µ ≥ 1.
(The particular case µ = 2, p = 3 leads to the KdV equation, see [15] and references therein.) In this case, formula (2) reads
(−cs − Λµ )φ +
φ p−1 = A, p−1
(11)
which implies that cs , A and the interval (−l, l) are not independent; specifically
l
φ dx +
− cs −l
l
−l
φ p−1 dx − 2lA = 0. p−1
(12)
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Fig. 1. Periodic traveling wave generation of (10) with A = cs = 1 and p = 3, µ = 0.8. (a) Approximate profiles; (b) Phase portrait; (c) Discrepancy (14) vs. number of iterations; (d) Residual error (13) vs. number of iterations. (Semi-logarithm scale in both cases.)
Relations of the form (12) determine the existence of the profiles (see e. g. Corollary 4.5 in [15]). Constant solutions φ = C of (11) (step (S1)) must satisfy C p−1 p−1
− cs C − A = 0,
A ∈ R,
while (4) takes the form
(cs + L − C L(C )
1
)ϕ − p−1
p−2
p−1 p−1 j
j =2
C p−1−j ϕ j = 0.
N (ϕ)
The behavior of the algorithm is illustrated here by taking A = cs = 1 and two values for µ and p: µ = 0.8, p = 3 and µ = 1.5, p = 4. In both cases, the convergence was accelerated by using the minimal polynomial extrapolation method (MPE), [33,34]. The approximate profile along with the corresponding phase portrait (computed by using the spectral approximation), are shown in Figs. 1(a), (b) and 2(a), (b), respectively. Figs. 1(a) and 2(a) were computed with l = 8 and 1024 Fourier discretization points. By approximating the corresponding integrals with the trapezoidal rule, the left hand side of (12) gives 2.842171E − 14 and 7.105427E − 14, respectively. Another way to check the accuracy of the computed profiles and the performance of the iteration is shown in Figs. 1(c),(d) and 2(c),(d). In each case and as function of the number of iterations, two types of errors are displayed:
• The residual error (in Euclidean norm) RES ν = ∥Lϕ [ν] − N (ϕ [ν] )∥,
ν = 0, 1, . . . .
(13)
• The discrepancy between the sequence of stabilizing factors sν = s(ϕ ) and (in case of convergence) its limit one [ν]
SFE ν = |sν − 1|,
ν = 0, 1, . . . .
(14)
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Fig. 2. Periodic traveling wave generation of (10) with A = cs = 1 and p = 4, µ = 1.2. (a) Approximate profiles; (b) Phase portrait; (c) Discrepancy (14) vs. number of iterations; (d) Residual error (13) vs. number of iterations. (Semi-logarithm scale in both cases.)
a
b
Fig. 3. Periodic traveling wave generation of (10) with A = cs = 1 and p = 3, µ = 0.8. (a) Peak amplitude error vs. time; (b) Speed error vs. time.
A final test of accuracy was made by taking the last computed profile as initial condition for the numerical integration of (9). The code to this end was similar to that performed in [13]. The experiments reveal an evolution of the initial profile without any relevant backward or forward disturbances. This is illustrated by Fig. 3(a),(b) which show, respectively, the evolution of the error between the peak amplitude (resp. speed) of the numerical approximation and that of the initial condition. (A similar figure for the peak-to-peak amplitude can be displayed.) The speed is computed by using (11) in the sense that if
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Fig. 4. (a), (b) Approximate periodic traveling wave profile of (10) and phase portrait with A = 0.1, cs = 0.08, l = 16 and p = 5, µ = 0.2. (c), (d) Approximate periodic traveling wave profile of (10) and phase portrait with A = 0.1, cs = 0.1, l = 8 and p = 5, µ = 0.2.
U n is the corresponding numerical solution at time tn then we can approximate cs ≈
1
⟨U n , U n ⟩
(U n )p−1 − A − Λµ U n , U n p−1
.
Once the accuracy is tested, the algorithm may be used to explore some additional properties of the waves, beyond the theoretical results. For (9) and by way of illustration, the code can generate numerically some periodic traveling wave profiles in cases that extend formula (10) for µ < 1. This is illustrated in Fig. 4, which shows some computed profiles corresponding to µ = 0.2 and a value of p larger than the limit pmax (0.2) = 5/2. 3.2. Extended Boussinesq system The second example is concerned with the so-called e-Boussinesq system
ηt = −d1 Wx − d2 Wxxx − d4 (W η)x + d5 (W η2 )x , Wt = −
1 d1
ηx − d3 Wxxt −
d4
(15)
(W 2 )x + d5 (W 2 η)x .
2
(16)
Eqs. (15), (16) appear as a model for the bidirectional propagation of an interfacial wave η(x, t ) between two fluid layers with W (x, t ) standing for the horizontal velocity of the flow. The constants are defined as d1 = d3 =
H r +H
κ d1 2
,
,
d2 =
H2 2(r + H )2 2
d4 =
H −r
(r + H )
2
,
κ + (1 + rH ) , 2
3
d5 =
r (1 + H )2
(r + H )3
,
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where r and H are, respectively, the (dimensionless) density and depth ratios while κ is related to the physical depth below and above the unperturbed interface and is taken within the range −(1 + rH ) ≤ κ ≤ − 23 (1 + rH ). The model is described in detail in [19], where the propagation and collision of solitary waves and fronts are also studied numerically. To our knowledge existence of periodic traveling wave solutions has not been analyzed (theoretically or numerically). Suggested here is some numerical evidence of such waves by using the method described in Section 2. System (15), (16) is of the form (1) with m = 2 and
L=
d1 + d2 ∂xx
0
−
1
,
0
d1
N =
0 0
0
d3 ∂xx
,
W η(d4 − d5 η)
f (η, W ) = 2 W
d4 2
− d5 η
.
In this case, periodic traveling wave solutions η = η(X ), W = W (X ), X = x − cs t of speed cs must satisfy the system for the profiles
−W η(−d4 + d5 η) −(d1 + d2 ∂xx ) A η + 1 , = d4 A2 W −W 2 − + d 5 η cs (1 + d3 ∂xx )
cs
−
1
(17)
2
d1
for some integration constants A1 , A2 . The search for constant solutions η = C1 , W = C2 of (17) leads to the algebraic system
−d 1
cs
1 −
C1
d4 C1 C2 − d5 C12 C2
= d4
C2
cs
d1
2
C22 − d5 C1 C22
+ A1 , A2
in such a way that if cs2 ̸= 1 then C2 =
cs C1 − A1 d1 + d4 C1 − d5 C12
,
while C1 must be a root of the polynomial P (z ) = −
1
z (d1 + d4 z − d5 z 2 )2 + cs (cs z − A1 )(d1 + d4 z − d5 z 2 )
d1
+
d5 z −
d4 2
(cs z − A1 )2 − A2 (d1 + d4 z − d5 z 2 )2 .
= W − C2 is of the form Finally, the system for the differences η = η − C1 , W
(−d4 + 2d5 C1 )C2 + d5 C22 +cs (1 + d3 ∂xx ) −(d1 + d2 ∂xx )
cs
−
1 d1
d5 C2 η +W η(−d4 + d5 η) + d5 W η 2
= −
2 W
−
d4 2
C1 (−d4 + d5 C1 )
2
2C2
η d4 W − + d5 C1 2
. 2 + d5 η + 2d5 C2 W η + d5 W η
(18)
and approximate periodic traveling wave profiles Then, as mentioned in Section 2, system (18) is iteratively solved for η, W + C2 . for the values A1 , A2 are obtained from the formulas η = η + C1 , W = W In [19] a maximum wave velocity vmax = 1 +
(H 2 − r )2 , 8rH (1 + H )2
is derived and solitary waves are characterized by wave velocities larger than one. We illustrate the performance of the method by generating periodic traveling waves with a speed close to vmax (specifically cs = vmax − 10−4 in the experiments below) and two different values for r , H and the constants A1 , A2 . (In both experiments κ = −(1 + rH ) has been fixed.) The first case, Fig. 5, corresponds to r = 0.8, H = 0.95, A1 = −1, A2 = −2. Fig. 5(a) shows the approximate η, W profiles (also obtained by using additional acceleration with the MPE method) and the periodic character is confirmed by the phase portrait of η in Fig. 5(b). The accuracy of the computation is again evaluated in Fig. 5(c) and (d) with the behavior of the errors (13) and (14) with respect to the number of iterations. The same information is provided in Fig. 6 for the case r = 0.8, H = 1.8, A1 = 1, A2 = 1. Note here that the periodic form of the approximate η profile is different and closer to the ‘table-top’ form of the solitary waves shown in [19]. This suggests the connection between these periodic profiles and some associated solitary wave in the limit of large wavelength, [3,5].
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Fig. 5. Periodic traveling wave generation of (17) with cs = vmax − 10−4 . A1 = −1, A2 = −2, r = 0.8, H = 0.95, s = −1 − rH. vmax = 1.000454. (a) Approximate profiles; (b) η phase portrait; (c) Residual error (13) vs. number of iterations; (d) Discrepancy (14) vs. number of iterations (semilogarithmic scale).
4. Concluding remarks In this paper an iterative technique to generate periodic traveling wave solutions of some nonlinear dispersive wave systems is proposed. The method is based on a suitable change of variable to deal with the presence of some constants of integration in the problem and the application of a fixed point iteration of Petviashvili type which can be applied when the nonlinearity contains homogeneous terms of different degree, [16]. This strategy overcomes two difficulties not totally solved in some previous algorithms, [13], for this purpose, namely:
• The presence of constants that prevents the use of Petviashvili type methods, which have been shown to be an alternative to compute solitary wave solutions for the models under study (see [11] and references therein).
• The presence of nonlinearities with homogeneous terms of different degree, for which the application of the Petviashvili method, by its own formulation, is not suitable. The scheme has been illustrated by computing periodic traveling waves in two problems of additional interest: the generalized fractional KdV equation and the extended Boussinesq system. In the first case, existence and stability of periodic traveling waves have been studied theoretically but, to our knowledge, no approximate profiles have been computed. The second example suggests the formation of periodic traveling waves in the model, for which there do not appear to be existence results. Acknowledgment This research has been supported by project MTM2014-54710-P.
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d
Fig. 6. Periodic traveling wave generation of (17). Approximate profiles and phase portraits with cs = vmax − 10−4 . A1 = 1, A2 = 1, r = 0.8, H = 1.8, s = −1 − rH. vmax = 1.065919. (a) Approximate profiles; (b) η phase portrait; (c) Residual error (13) vs. number of iterations; (d) Discrepancy (14) vs. number of iterations (semi-logarithmic scale).
References [1] L. Abdelouhab, J.L. Bona, M. Felland, J.-C. Saut, Non-local models for nonlinear dispersive waves, Physica D 40 (1989) 360–392. [2] J.L. Bona, On solitary waves and their role in the evolution of long waves, in: H. Amann, N. Bazley, K. Kirchgässner (Eds.), Applications of Nonlinear Analysis in the Physical Sciences, Pitman, London, 1981, pp. 183–205. [3] J.L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res. 37 (1981) 21–30. [4] J. Angulo Pava, Nonlinear Dispersive Equations, in: Mathematical Surveys and Monographs, vol. 156, American Math. Soc., Providence, RI, 2009. [5] H. Chen, J.L. Bona, Periodic traveling wave solutions of nonlinear dispersive evolution equations, Discrete Contin. Dyn. Syst. Ser. B 33 (11& 12) (2013) 4841–4873. [6] H. Chen, Existence of periodic traveling-wave solutions of nonlinear, dispersive wave equations, Nonlinearity 17 (2004) 2041–2056. [7] H. Chen, M. Chen, N.V. Nguyen, Cnoidal wave solutions to Boussinesq systems, Nonlinearity 20 (2007) 1443–1461. [8] V. Mikyoung Hur, M.A. Johnson, Stability of periodic traveling waves for nonlinear dispersive equations, SIAM J. Math. Anal. 47 (5) (2015) 3528–3554. [9] F.A. Pipicano, J.C. Muñoz Grajales, Existence of periodic traveling wave solutions for a regularized Benjamin–Ono system, J. Differential Equations 259 (2015) 7503–7528. [10] J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover Publications, New York, 2000. [11] J. Alvarez, A. Duran, Petviashvili type methods for traveling wave computations: I. Analysis of convergence, J. Comput. Appl. Math. 266 (2014) 39–51. [12] J.P. Boyd, Why Newton’s method is hard for travelling waves: Small denominators, KAM theory, Arnold’s linear Fourier problem, non-uniqueness, constraints and erratic failure, Math. Comput. Simul. 74 (2007) 72–81. [13] J. Alvarez, A. Duran, A numerical scheme for periodic travelling-wave simulations in some nonlinear dispersive wave models, J. Comput. Appl. Math. 235 (2011) 1790–1797. [14] V.I. Petviashvili, Equation of an extraordinary soliton, Soviet J. Plasma Phys. 2 (1976) 257–258. [15] J. Angulo Pava, J.L. Bona, M. Scialom, Stability of cnoidal waves, Adv. Differential Equations 11 (2006) 1321–1374. [16] J. Alvarez, A. Duran, An extended Petviashvili method for the numerical generation of traveling and localized waves, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 2272–2283. [17] J. Alvarez, A. Duran, On Petviashvili type methods for traveling wave computations: II. Acceleration with vector extrapolation methods, Math. Comput. Simul. 123 (2016) 19–36. [18] M.A. Johnson, Stability of small periodic waves in fractional KdV-type equations, SIAM J. Math. Anal. 45 (5) (2013) 3168–3193. [19] H.Y. Nguyen, F. Dias, A Boussinesq system for two-way propagation of interfacial waves, Physica D 237 (2008) 2365–2389. [20] J.L. Bona, M. Chen, J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and linear theory, J. Nonlinear Sci. 12 (2002) 283–318. [21] J.L. Bona, M. Chen, J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory, Nonlinearity 17 (2004) 925–952.
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[22] F. Serre, Contribution à l’étude des écoulements permanents et variables dans les canaux, Houille Blanche 8 (1953) 374–388. [23] F. Serre, Contribution à l’étude des écoulements permanents et variables dans les canaux, Houille Blanche 8 (1953) 830–872. [24] A. Leykin, J. Verschelde, D. Zao, Newton’s method with deflation for isolated singularities of polynomial systems, Theoret. Comput. Sci. 359 (1–3) (2006) 111–122. [25] S. Kiefer, M. Luttenberger, J. Esparza, On the convergence of Newton’s method for monotone systems of polynomial equations, in: Proceedings of the 39th ACM Symposium on Theory of Computing STOC07, 2007, pp. 217–226. [26] G.H. Golub, C.F. Van Loan, Matrix Computations, third ed., Johns Hopkins Univ. Press, Baltimore, Maryland, 1996. [27] D. Bondyfalat, B. Mourrain, V.Y. Pan, Controlled iterative methods for solving polynomial systems, in: Proceedings of the 1998 International Minisymposium on Symbolic and Algebraic Computation ISSAC98, 1998, pp. 252–259. [28] C.B. Garcia, W.I. Zangwill, Finding all solutions to polynomial systems and other systems of equations, Math. Program. 16 (1979) 159–176. [29] J. Verschelde, Homotopy Methods for Solving Polynomial Systems, tutorial at ISSAC2005, July 24–27, 2005, Key Laboratory of Mathematics Mechanization, Beijing, China. [30] D. Lazard, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, in: Proceeedings of the European Computer Algebra Conference on Computer Algebra, 1983, pp. 146–156. [31] B.L. Van der Waerden, Algebra, Vol. 1, Springer-Verlag, New York, 1991. [32] W.T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, in: W. Bledsoe, D. Loveland (Eds.), Automated Theorem proving: After 25 Years, AMS, 1983, pp. 213–234. [33] C. Brezinski, M. Redivo Zaglia, Extrapolation Methods, Theory and Practice, North-Holland, Amsterdam, 1991. [34] A. Sidi, Practical Extrapolation Methods, Theory and Applications, Cambridge University Press, New York, 2003.
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Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems J. Álvarez a,b , A. Durán c,b,∗ a
Department of Applied Mathematics, University of Valladolid, Paseo del Cauce 59, 47011, Valladolid, Spain
b
IMUVA, Institute of Mathematics of University of Valladolid, Spain
c
Department of Applied Mathematics, University of Valladolid, Paseo de Belen 15, 47011, Valladolid, Spain
highlights • We propose a method to compute periodic traveling waves. • The method can be applied to some nonlinear dispersive systems. • It is checked in fractional KdV equations and Boussinesq systems.
article
abstract
info
Article history: Received 5 November 2015 Received in revised form 26 July 2016
In this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced. The method is based on a suitable modification of a fixed point algorithm of Petviashvili type and solves several drawbacks of some previous algorithms presented in the literature. By way of illustration, the method is applied to generate numerically periodic traveling waves of two problems of interest: the fractional KdV type equations and the extended Boussinesq system. © 2016 Elsevier B.V. All rights reserved.
MSC: 65H10 65M99 35C99 35C07 76B25 Keywords: Nonlinear dispersive equations Periodic traveling waves Iterative methods for nonlinear systems Petviashvili type methods Fractional KdV equation e-Boussinesq system
1. Introduction Many mathematical models for nonlinear dispersive wave propagation are governed by evolution systems of the form ut + Lux + f (u)x − Nut = 0,
x ∈ R, t ≥ 0.
(1)
In (1) u = u(x, t ) = (u1 (x, t ), . . . , um (x, t )) is a real-valued vector function (m ≥ 1) of x ∈ R, t ≥ 0, L and N are m × m matrices with entries given by Fourier multiplier operators with symbols lij (ξ ), nij (ξ ), i, j = 1, . . . , m, ξ ∈ R T
∗
Corresponding author at: Department of Applied Mathematics, University of Valladolid, Paseo de Belen 15, 47011, Valladolid, Spain. E-mail addresses: [email protected] (J. Álvarez), [email protected] (A. Durán).
http://dx.doi.org/10.1016/j.cam.2016.08.037 0377-0427/© 2016 Elsevier B.V. All rights reserved.
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respectively and f = (f1 , . . . , fm )T : Rm → Rm is smooth (see e. g. [1] and references therein). Much of the relevance of (1) is due to the possible existence of traveling wave solutions, [2]. They are characterized as waves of permanent form uj (x, t ) = φj (x − cs t ), j = 1, . . . , m, traveling with fixed speed cs ∈ R. In the study of existence and dynamics of such solutions, numerical computation plays a relevant role since typically the profiles φj are not known to exist or when the problem of existence is overcome, there are no exact formulas. This paper is concerned with the numerical generation of traveling wave solutions of (1) of periodic type, that is where the profiles φj are periodic of some period 2l, l > 0. The vector φ = (φ1 , . . . , φm ) must therefore satisfy the system
(−cs (I − N ) + L)φ + f (φ) = A,
(2)
where A = (A1 , . . . , Am ) is (nonzero in general) constant. (Note that, contrary to (1), system (2) is stationary since it only depends on the variable x − cs t.) The existence and dynamics of these solutions are relevant in their own and because of the relation with solitary wave solutions in the long-wavelength limit, [3]. Results on existence and stability for particular cases of (1) can be seen in the literature, see e. g. [4] and references therein. It is also worth mentioning some additional recent results. Thus, by way of modest illustration, Chen and Bona, [5], establish existence of periodic traveling wave solutions of (1) for the case m = 1, N = 0, under some hypotheses on f and L as well as stability under perturbations of the same period. (This study generalizes previous works on this matter, [6].) Also, Chen et al., [7], analyze the existence of periodic traveling waves of systems of Boussinesq type. On the other hand, Mikyoung Hur and Johnson, [8], study periodic traveling waves for KdV equations with fractional dispersion, analysis that may be generalized to models of the form (1) with m = 1, N = 0. Finally, Pipicano and Muñoz, [9], discuss the existence of these waves for a regularized Benjamin–Ono system. As far as the numerical generation is concerned, the literature emphasizes the use of pseudospectral Fourier approximation to (2) and different resolution of the resulting discrete equations, from the application of a Fourier–Galerkin algorithm as explicit analytical approximation, [10], to the use of iterative methods. Concerning this last approach, the application of classical methods like the fixed point algorithm is in general not convergent, [11], while the Newton method (used e. g. in [9]) suffers from important drawbacks, [12]. In this sense, in [13] a numerical scheme for the computation of periodic traveling waves of (1) with m = 1, N = 0 was proposed. This was based on a Fourier pseudospectral discretization in space of (1) along with time integration with conservation properties of invariant quantities of the periodic boundary-value problem associated to (1). The periodic traveling wave profile was approximated by using the iterative Petviashvili method, [14,11]. The resulting scheme ensured an accurate computation of the profile and speed of the waves. The present work is a continuation of [13] and attempts to complete it in several points. The first one is the extension of the study to systems of the form (1). The second one concerns the feature that the generation of the profile may involve nonzero constants A in (2) which make the Petviashvili method unable to be used. A theoretical strategy of reducing (2) to the case A = 0, considered in e. g. [15,5], forces to make a change in the nonlinearity which also prevents the application of this iterative scheme. Since homogeneity of degree greater than one in magnitude is required in [11] and the presence of constants is not considered in [16], the present paper complements in this sense the previous when applied to a new problem of numerical generation of periodic traveling waves. Indeed, the case of zero constant reduces to consider the method proposed in [16] and the case of zero constant and homogeneous (of degree greater than one in magnitude) nonlinear term deals with the application of the Petviashvili method to compute periodic traveling waves in the nonlinear dispersive wave equations under study. In order to overcome these two last drawbacks, the present paper introduces a two-fold strategy. This is based on dividing the solution into a constant plus a second profile. The constant is a solution of a polynomial system, which is iteratively solved, while the second profile can be approximated by a fixed point algorithm of Petviashvili type complemented by acceleration techniques to improve the performance, [17]. The structure of the paper is as follows. In Section 2 the proposed algorithm is described under some hypotheses on (1). The performance of the method is tested in Section 3 by considering the computation of periodic traveling waves in two models. They are the generalized fractional KdV equation, [1,18], and the e-Boussinesq system, [19]. In the first case, there are theoretical results of existence of waves but no exact formulas for the profiles. In the second case and to our knowledge, the problem of existence has not been studied yet. For this reason, the existence of such waves is implicitly assumed throughout the paper although, since the theoretical question is out of our goal, no specific hypotheses will be posed (see the above comments and mentioned references in this Introduction about this problem). Finally, Section 4 contains some concluding remarks. 2. Preliminaries and numerical method The numerical technique below is applied to compute periodic traveling wave solutions of (1) under two assumptions: (H1) For each (i, j), 1 ≤ i, j ≤ m the Fourier symbols lij and nij of the (i, j)th entries of the matrix operators L and N are real, even and continuous. (H2) For each j = 1, . . . , m, the nonlinear term fj is a polynomial in Rm with fj (0) = ∂∂x fj (0) = 0, k = 1, . . . , m and degree k pj − 1 with pj ≥ 3. As will be seen below, conditions (H1), (H2) are required for the derivation of the proposed numerical resolution of (2): while (H1) implies that any constant vector is in the kernel of the operators L and N, the use of (H2) allows to transform (2) into a
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new system where the nonlinear term only contains homogeneities of degree at least two. On the other hand, many of the wave propagation models governed by systems (1) presented in the literature involve linear and nonlinear terms satisfying (H1) and (H2) respectively. By way of illustration, scalar cases like the KdV, BBM, Benjamin–Ono and Benjamin equations along with their generalized versions (see e. g. [1,4] and references therein) or systems like the four-parameter Boussinesq equations, [20,21], belong to this family. On the contrary, important models like Serre equations, [22,23], are out of it. The method proposed here for the numerical resolution of (2) consists of the following steps: (S1) Finding (approximate or exact) real constant solutions φ = C = (C1 , . . . , Cm ) of (2) that is
(−cs (I − N ) + L)C + f (C ) = A.
(3)
(S2) The change of variables ϕ = φ − C , when φ is a solution of (2) and (3) lead to a system for ϕ of the form:
L(C )ϕ − N (ϕ) = 0,
(4)
with
L(C ) = cs − (L + cs N ) − f ′ (C ), N (ϕ) = (N1 (ϕ), . . . , Nm (ϕ))T Nk (ϕ) =
p−1 m 1 Djxi ,...,xi fk (C )ϕxi · · · ϕxi j 1 j 1 j ! i ,...,i =1 j =2 j
1
where p = max{p1 , . . . , pm } and Djxi ,...,xi fk (C ) = j 1
∂ j fk (C ), ∂ xi1 · · · ∂ xij
k = 1, . . . , m
(S3) System (4) is iteratively solved with an extended version of the Petviashvili method, [16]. The nonlinear term N is rewritten in the form
N (ϕ)
=
p−1
N (j) (ϕ),
(j)
N (j) (ϕ) = (N1 (ϕ), . . . , Nm(j) (ϕ))T
j =2
(j)
Nk (ϕ) =
m
1
j! i ,...,i =1 j 1
Djxi ,...,xi fk (C )ϕxi · · · ϕxi , 1 j j 1
k = 1, . . . , m.
Now each vector N (j) (ϕ) is homogeneous of degree j = 2, . . . , p − 1. Thus, from ϕ [0] ̸= 0 and for ν = 0, 1, . . . the following iteration is considered: p−1
L(C )ϕ [ν+1] =
sj (ϕ [ν] )N (j) (ϕ [ν] ),
ν = 0, 1, . . . ,
(5)
j =2
where sj (u) = s(u)αj ,
αj =
j j−1
,
s(u) =
⟨L(C )u, u⟩ , ⟨N (u), u⟩
(6)
j = 2, . . . , p − 1.
The so-called stabilizing factors sj in (6) are obtained by using the same quotient s(u) (where ⟨·, ·⟩ stands for the usual Euclidean inner product) but changing the exponent αj . (Note in particular that in the case of convergence, s(ϕ [ν] ) → 1.) Formulas (5) and (6) correspond to an extension of the classical Petviashvili scheme, [14], when the nonlinearity consists of several homogeneous terms with different degree of homogeneity. The procedure considered here takes advantage of the behavior of the Petviashvili method and its generalization for nonlinear systems with homogeneous nonlinearities of degree greater than one in magnitude. For such systems, the degree of homogeneity is an eigenvalue of the iteration matrix at the fixed point for the classical fixed point algorithm, which in general enables it to be convergent. The Petviashvili iteration acts like a filter of this harmful eigenvalue, in such a way that its corresponding iteration matrix at the fixed point transforms the degree of homogeneity to an eigenvalue of magnitude below one and preserves the rest of the spectrum of the iteration matrix of the classical fixed point algorithm, translating then the question of the convergence to this remaining part of the spectrum, [11]. This behavior was also experimentally observed in the extended version proposed in [16] for nonlinearities consisting of superposition of homogeneous terms of degree above one in magnitude. The polynomial system (3) can be iteratively solved by any of the algorithms developed in the literature to this end: for instance, Newton-type iterations, [24,25] (since in this case the drawbacks mentioned in [12] do not apply), iterative
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methods, [26,27], homotopy continuation, [28,29] or symbolic computational procedures like Gröbner bases, [30], resultant elimination methods, [31] or Wu’s method, [32]. The operator (2) taking part of L(C ) in (4) suggests to formulate (5), (6) in Fourier space for the Fourier coefficients [ν] , n ∈ Z of the iterations ϕ [ν] , ν = 0, 1, . . . Then for fixed c and l > 0 the problem (2) on (−l, l) with periodic boundary ϕ s n conditions is considered and (5), (6) take the form
cs I − f ′ (C ) + ( L + cs N)
nπ l
[ν+1] = ϕ n
cs I − f ′ (C ) + ( L + cs N)
n∈Z
sj (ϕ [ν] ) N (j) (ϕ [ν] ) n ,
n ∈ Z, ν = 0, 1, . . . ,
(7)
j=2
sj (u) =
p−1
nπ l
un , un
αj
un N (j) (u) n
,
j = 2, . . . , p − 1,
(8)
n∈Z
N )( nlπ ) denotes the Fourier symbol matrix with entries (lij ( nlπ ) + cs nij ( nlπ )), i, j = 1, . . . , m, αj is given by (6) where ( L + cs and un = (( u1 )n , . . . , ( um )n )T , n ∈ Z. In the experiments below, system (4) is discretized on an interval with a Fourier pseudospectral method as in [13], see [10]. This allows to implement the iteration by using the discrete Fourier version of (7), (8) and FFT techniques. Furthermore, in some of the experiments, the resulting scheme has been implemented with vector extrapolation methods to accelerate the convergence. The literature on these techniques is extensive, see e. g. [33,34] and references therein. (The use of vector extrapolation methods in traveling wave simulations is analyzed in [17].) 3. Numerical examples This section is devoted to illustrate the behavior of the algorithm to generate numerically periodic traveling waves. This will be shown by considering two models of the form (1): a generalized version of the fractional KdV equation and the extended Boussinesq system. Besides these illustrative purposes, there are additional reasons to justify the choice of these equations. In the first case, periodic traveling wave solutions are known to exist (under certain hypotheses) but without exact formulas for the profiles. Then the algorithm may be useful to compute accurate approximations, in order to be used, for example, in further analyses, by numerical means, of the dynamics. In the case of the second system, to our knowledge, the problem of existence of periodic traveling wave solutions has not been studied yet and the proposed algorithm may additionally serve as a preliminary numerical evidence of such waves in the model. 3.1. Generalized fractional KdV equation The first example treated here concerns the generation of periodic traveling waves of KdV models with fractional dispersion, of the form ut − Λµ ux + (f (u))x = 0,
(9)
where
v(ξ ) = |ξ | Λ v(ξ ),
0 < µ ≤ 2,
f (u) =
up−1 p−1
,
p ≥ 3.
Eq. (9) is of the form (1) with m = 1, N = 0, L = Λµ . Existence and stability of periodic traveling wave solutions are analyzed in [8,18] by using variational arguments. Specifically, existence is obtained for 0 < p < pmax where pmax = pmax (µ) :=
2 1−µ
+∞
if µ < 1
(10)
if µ ≥ 1.
(The particular case µ = 2, p = 3 leads to the KdV equation, see [15] and references therein.) In this case, formula (2) reads
(−cs − Λµ )φ +
φ p−1 = A, p−1
(11)
which implies that cs , A and the interval (−l, l) are not independent; specifically
l
φ dx +
− cs −l
l
−l
φ p−1 dx − 2lA = 0. p−1
(12)
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Fig. 1. Periodic traveling wave generation of (10) with A = cs = 1 and p = 3, µ = 0.8. (a) Approximate profiles; (b) Phase portrait; (c) Discrepancy (14) vs. number of iterations; (d) Residual error (13) vs. number of iterations. (Semi-logarithm scale in both cases.)
Relations of the form (12) determine the existence of the profiles (see e. g. Corollary 4.5 in [15]). Constant solutions φ = C of (11) (step (S1)) must satisfy C p−1 p−1
− cs C − A = 0,
A ∈ R,
while (4) takes the form
(cs + L − C L(C )
1
)ϕ − p−1
p−2
p−1 p−1 j
j =2
C p−1−j ϕ j = 0.
N (ϕ)
The behavior of the algorithm is illustrated here by taking A = cs = 1 and two values for µ and p: µ = 0.8, p = 3 and µ = 1.5, p = 4. In both cases, the convergence was accelerated by using the minimal polynomial extrapolation method (MPE), [33,34]. The approximate profile along with the corresponding phase portrait (computed by using the spectral approximation), are shown in Figs. 1(a), (b) and 2(a), (b), respectively. Figs. 1(a) and 2(a) were computed with l = 8 and 1024 Fourier discretization points. By approximating the corresponding integrals with the trapezoidal rule, the left hand side of (12) gives 2.842171E − 14 and 7.105427E − 14, respectively. Another way to check the accuracy of the computed profiles and the performance of the iteration is shown in Figs. 1(c),(d) and 2(c),(d). In each case and as function of the number of iterations, two types of errors are displayed:
• The residual error (in Euclidean norm) RES ν = ∥Lϕ [ν] − N (ϕ [ν] )∥,
ν = 0, 1, . . . .
(13)
• The discrepancy between the sequence of stabilizing factors sν = s(ϕ ) and (in case of convergence) its limit one [ν]
SFE ν = |sν − 1|,
ν = 0, 1, . . . .
(14)
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Fig. 2. Periodic traveling wave generation of (10) with A = cs = 1 and p = 4, µ = 1.2. (a) Approximate profiles; (b) Phase portrait; (c) Discrepancy (14) vs. number of iterations; (d) Residual error (13) vs. number of iterations. (Semi-logarithm scale in both cases.)
a
b
Fig. 3. Periodic traveling wave generation of (10) with A = cs = 1 and p = 3, µ = 0.8. (a) Peak amplitude error vs. time; (b) Speed error vs. time.
A final test of accuracy was made by taking the last computed profile as initial condition for the numerical integration of (9). The code to this end was similar to that performed in [13]. The experiments reveal an evolution of the initial profile without any relevant backward or forward disturbances. This is illustrated by Fig. 3(a),(b) which show, respectively, the evolution of the error between the peak amplitude (resp. speed) of the numerical approximation and that of the initial condition. (A similar figure for the peak-to-peak amplitude can be displayed.) The speed is computed by using (11) in the sense that if
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Fig. 4. (a), (b) Approximate periodic traveling wave profile of (10) and phase portrait with A = 0.1, cs = 0.08, l = 16 and p = 5, µ = 0.2. (c), (d) Approximate periodic traveling wave profile of (10) and phase portrait with A = 0.1, cs = 0.1, l = 8 and p = 5, µ = 0.2.
U n is the corresponding numerical solution at time tn then we can approximate cs ≈
1
⟨U n , U n ⟩
(U n )p−1 − A − Λµ U n , U n p−1
.
Once the accuracy is tested, the algorithm may be used to explore some additional properties of the waves, beyond the theoretical results. For (9) and by way of illustration, the code can generate numerically some periodic traveling wave profiles in cases that extend formula (10) for µ < 1. This is illustrated in Fig. 4, which shows some computed profiles corresponding to µ = 0.2 and a value of p larger than the limit pmax (0.2) = 5/2. 3.2. Extended Boussinesq system The second example is concerned with the so-called e-Boussinesq system
ηt = −d1 Wx − d2 Wxxx − d4 (W η)x + d5 (W η2 )x , Wt = −
1 d1
ηx − d3 Wxxt −
d4
(15)
(W 2 )x + d5 (W 2 η)x .
2
(16)
Eqs. (15), (16) appear as a model for the bidirectional propagation of an interfacial wave η(x, t ) between two fluid layers with W (x, t ) standing for the horizontal velocity of the flow. The constants are defined as d1 = d3 =
H r +H
κ d1 2
,
,
d2 =
H2 2(r + H )2 2
d4 =
H −r
(r + H )
2
,
κ + (1 + rH ) , 2
3
d5 =
r (1 + H )2
(r + H )3
,
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where r and H are, respectively, the (dimensionless) density and depth ratios while κ is related to the physical depth below and above the unperturbed interface and is taken within the range −(1 + rH ) ≤ κ ≤ − 23 (1 + rH ). The model is described in detail in [19], where the propagation and collision of solitary waves and fronts are also studied numerically. To our knowledge existence of periodic traveling wave solutions has not been analyzed (theoretically or numerically). Suggested here is some numerical evidence of such waves by using the method described in Section 2. System (15), (16) is of the form (1) with m = 2 and
L=
d1 + d2 ∂xx
0
−
1
,
0
d1
N =
0 0
0
d3 ∂xx
,
W η(d4 − d5 η)
f (η, W ) = 2 W
d4 2
− d5 η
.
In this case, periodic traveling wave solutions η = η(X ), W = W (X ), X = x − cs t of speed cs must satisfy the system for the profiles
−W η(−d4 + d5 η) −(d1 + d2 ∂xx ) A η + 1 , = d4 A2 W −W 2 − + d 5 η cs (1 + d3 ∂xx )
cs
−
1
(17)
2
d1
for some integration constants A1 , A2 . The search for constant solutions η = C1 , W = C2 of (17) leads to the algebraic system
−d 1
cs
1 −
C1
d4 C1 C2 − d5 C12 C2
= d4
C2
cs
d1
2
C22 − d5 C1 C22
+ A1 , A2
in such a way that if cs2 ̸= 1 then C2 =
cs C1 − A1 d1 + d4 C1 − d5 C12
,
while C1 must be a root of the polynomial P (z ) = −
1
z (d1 + d4 z − d5 z 2 )2 + cs (cs z − A1 )(d1 + d4 z − d5 z 2 )
d1
+
d5 z −
d4 2
(cs z − A1 )2 − A2 (d1 + d4 z − d5 z 2 )2 .
= W − C2 is of the form Finally, the system for the differences η = η − C1 , W
(−d4 + 2d5 C1 )C2 + d5 C22 +cs (1 + d3 ∂xx ) −(d1 + d2 ∂xx )
cs
−
1 d1
d5 C2 η +W η(−d4 + d5 η) + d5 W η 2
= −
2 W
−
d4 2
C1 (−d4 + d5 C1 )
2
2C2
η d4 W − + d5 C1 2
. 2 + d5 η + 2d5 C2 W η + d5 W η
(18)
and approximate periodic traveling wave profiles Then, as mentioned in Section 2, system (18) is iteratively solved for η, W + C2 . for the values A1 , A2 are obtained from the formulas η = η + C1 , W = W In [19] a maximum wave velocity vmax = 1 +
(H 2 − r )2 , 8rH (1 + H )2
is derived and solitary waves are characterized by wave velocities larger than one. We illustrate the performance of the method by generating periodic traveling waves with a speed close to vmax (specifically cs = vmax − 10−4 in the experiments below) and two different values for r , H and the constants A1 , A2 . (In both experiments κ = −(1 + rH ) has been fixed.) The first case, Fig. 5, corresponds to r = 0.8, H = 0.95, A1 = −1, A2 = −2. Fig. 5(a) shows the approximate η, W profiles (also obtained by using additional acceleration with the MPE method) and the periodic character is confirmed by the phase portrait of η in Fig. 5(b). The accuracy of the computation is again evaluated in Fig. 5(c) and (d) with the behavior of the errors (13) and (14) with respect to the number of iterations. The same information is provided in Fig. 6 for the case r = 0.8, H = 1.8, A1 = 1, A2 = 1. Note here that the periodic form of the approximate η profile is different and closer to the ‘table-top’ form of the solitary waves shown in [19]. This suggests the connection between these periodic profiles and some associated solitary wave in the limit of large wavelength, [3,5].
J. Álvarez, A. Durán / Journal of Computational and Applied Mathematics (
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Fig. 5. Periodic traveling wave generation of (17) with cs = vmax − 10−4 . A1 = −1, A2 = −2, r = 0.8, H = 0.95, s = −1 − rH. vmax = 1.000454. (a) Approximate profiles; (b) η phase portrait; (c) Residual error (13) vs. number of iterations; (d) Discrepancy (14) vs. number of iterations (semilogarithmic scale).
4. Concluding remarks In this paper an iterative technique to generate periodic traveling wave solutions of some nonlinear dispersive wave systems is proposed. The method is based on a suitable change of variable to deal with the presence of some constants of integration in the problem and the application of a fixed point iteration of Petviashvili type which can be applied when the nonlinearity contains homogeneous terms of different degree, [16]. This strategy overcomes two difficulties not totally solved in some previous algorithms, [13], for this purpose, namely:
• The presence of constants that prevents the use of Petviashvili type methods, which have been shown to be an alternative to compute solitary wave solutions for the models under study (see [11] and references therein).
• The presence of nonlinearities with homogeneous terms of different degree, for which the application of the Petviashvili method, by its own formulation, is not suitable. The scheme has been illustrated by computing periodic traveling waves in two problems of additional interest: the generalized fractional KdV equation and the extended Boussinesq system. In the first case, existence and stability of periodic traveling waves have been studied theoretically but, to our knowledge, no approximate profiles have been computed. The second example suggests the formation of periodic traveling waves in the model, for which there do not appear to be existence results. Acknowledgment This research has been supported by project MTM2014-54710-P.
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Fig. 6. Periodic traveling wave generation of (17). Approximate profiles and phase portraits with cs = vmax − 10−4 . A1 = 1, A2 = 1, r = 0.8, H = 1.8, s = −1 − rH. vmax = 1.065919. (a) Approximate profiles; (b) η phase portrait; (c) Residual error (13) vs. number of iterations; (d) Discrepancy (14) vs. number of iterations (semi-logarithmic scale).
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