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Spatial chaos in a fourth-order nonlinear parabolic equation
1 October 2001
Physics Letters A 288 (2001) 299–304 www.elsevier.com/locate/pla
Spatial chaos in a fourth-order nonlinear parabolic equation S. Albeverio a , I.L. Nizhnik b,∗ a Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53 115 Bonn 1, Germany b Institute of Mathematics, 3 Tereschenkivska Str., Kiev 01 601, Ukraine
Received 14 March 2001; received in revised form 26 June 2001; accepted 8 August 2001 Communicated by A.P. Fordy
Abstract Bounded stationary solutions for a fourth-order extended Fisher–Kolmogorov equation with cubic-like piecewise linear nonlinearity are given. Those solutions which have a finite number of zeros are characterized by means of a set of integers associated with the distance between the zeros. The value of the spatial entropy is estimated and the existence of spatial chaos is shown. 2001 Elsevier Science B.V. All rights reserved. AMS classification: 35C05; 35G25; 35K57; 35K60; 58F13 Keywords: Diffusion equation; Entropy; Spatial chaos
1. Introduction In this Letter, we will be concerned with stationary solutions of bistable systems described by the extended Fisher–Kolmogorov equation ∂u ∂ 4u ∂ 2u (1) = −γ 4 + 2 − f (u), γ > 0, ∂t ∂x ∂x where the function f will be specified below as a cubic-like piecewise linear nonlinear one. Eq. (1) where f (u) = u3 − u was proposed [5,6] as a generalization of the classical Fisher–Kolmogorov equation ∂u ∂ 2 u = 2 + u − u3 . (2) ∂t ∂x The latter equation has a bistable nonlinearity in the sense that the ordinary differential equation du/dt = u − u3 has two stable stationary solutions, u = ±1. * Corresponding author.
E-mail address: [email protected] (I.L. Nizhnik).
It plays an important role in various areas of physics, chemistry, and biology [1,12,17]. The presence of a bistable nonlinearity in the diffusion equation is the cause for the appearance of various interesting phenomena such as self-organization, pattern formation and spatial chaos, see, e.g., [2–4,8–11, 13] and references therein. It was shown in [11,14,15] that for a difference analogue of Eq. (2), ∂un = d[un+1 + un−1 − 2un ] + un − u3n , (3) ∂t with a small value of the cell coupling constant d 0, stable stationary solutions un are close to the signature function σn that takes values +1 or −1 for each n such that |un − σn | < 4d. Here, every stable stationary solution un is uniquely determined by its skeleton, uˆ n = sign un , and an arbitrary signature function is the skeleton of a stationary stable solution. This result extends to the case of
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 5 3 2 - 1
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S. Albeverio, I.L. Nizhnik / Physics Letters A 288 (2001) 299–304
Eq. (3) with an arbitrary value of d and the piecewise linear cubic-like nonlinearity [15]: u − 1, u > 0, f (u) = u + 1, u < 0, (4) 0, u = 0. Nonlinearity (4) is also a bistable nonlinearity. It reproduces features of a cubic nonlinearity, and allows to obtain exact results in terms of explicit formulas. In particular, an estimate for the spatial entropy was obtained for Eq. (3) with nonlinearity (4). It has also been proved that there is a spatial chaos in system (3) [15]. For Eq. (2), which is a continuous analogue of Eq. (3), all bounded stationary solutions can be explicitly expressed in terms of elliptic functions and are reduced to periodic solutions or kinks. Thus bounded stationary solutions of Eq. (2) do not have chaotic behavior, as opposed to solutions of the discrete analogue equation (3). In this Letter, we prove that there is an effect of spatial chaos appearing in solutions of the fourthorder equation (1) with the piecewise linear cubic-like nonlinearity (4). Our method is based on the study of the location of zeros of stationary solutions of (1). This constructive approach is fundamentally different from the analysis of the qualitative behavior of solutions of the stationary equation (1) with a cubic nonlinearity which is given in [16].
We will study bounded stationary solutions of the extended Fisher–Kolmogorov equation (1) with the piecewise linear nonlinearity (4): 1 γ> . (5) 4 By definition we set sign 0 = 0. If y is a solution of Eq. (5), then −y is also a solution of this equation. Hence, we will be looking for a solution up to the sign. The linear equation corresponding to (5), γ y (4) − y
+ y = sign y,
γ y (4) − y
+ y = 0,
(6)
has four linearly independent solutions:
eαx sin βx.
e−αx sin βx,
+∞ y(x) = G(x − s) sign y(s) ds,
(9)
−∞
where G is explicitly expressed via solution (7) of the homogeneous equation (6): 1 G(x) = √ e−α|x| cos βx + λ sin β|x| . (10) 4α γ Remark 1. From representation (9) and expression (10) one has the a priori estimate for the bounded solution of Eq. (5), y(x)
+∞ G(s) ds
−∞
√ 4 γ √ , 1+2 γ
because of | cos βx| 1 and | sin βx| β|x|. By using representation (9), it is easy to get the following lemma.
2. Stationary solutions
e−αx cos βx,
The numbers α > 0 and β > 0 are found in terms of the parameter γ from (6) by 1 1 1/2 1 1/2 1 + − , β = , α= √ √ 2 γ 4γ 2 γ 4γ α λ = > 1. (8) β Bounded solutions of Eq. (5), for a given function sign y, can be represented in terms of the Green’s function G in the form of well-defined Lebesgue integral:
eαx cos βx, (7)
Lemma 1. Let y ≡ 0 be a bounded solution of Eq. (5) not changing sign on an interval [x0 − l, x0 + l]. Then, for all x ∈ [x0 − l, x0 + l] save for, possibly, a finite number of points, the function sign y(x) takes a constant value σ = ±1. We also have the following estimate: ∞ y(x0) − σ 4 G(s) ds < 2(1 + λ)e−λβl . (11) l
Remark 2. Let y be a bounded solution of Eq. (5) and let the distance l between a point x and the closest zero of the solution y satisfy the inequality βl ln(4/), 0 < 1. Then y(x) − sign y(x) < . (12)
S. Albeverio, I.L. Nizhnik / Physics Letters A 288 (2001) 299–304
Remark 3. For βl > ln 4, it follows from inequality (11) that |y(x0 ) − σ | < 1 and, hence, y(x0 ) = 0. Thus, if the distance l between neighboring zeros of a bounded solution y of Eq. (5) satisfies l > (1/β) ln 4, then the solution changes sign in a neighborhood of each zero. Theorem 1. Let {xk } be the increasingly ordered set of zeros of a bounded solution y = y(x, {xk }) of Eq. (5). Let a solution change its sign in a neighborhood of each zero xn . Then this solution is uniquely defined (up to the sign) by its zeros. Proof. By using the zeros {xk } of the solution y, it is possible to uniquely (up to the sign) construct the function sign y(x), without loss of generality assuming that 1, x2n < x < x2n+1 , sign y(x) = (13) −1, x2n+1 < x < x2n+2 , otherwise instead of y(x) we take −y(x). If the set of the zeros {xn } in (13) is bounded from below (above), it will be augmented by −∞ (+∞). Substitution of (13) into (9) gives an explicit expression for the solution y of Eq. (5) in terms of its zeros {xk } in the form of the absolutely convergent series: y(x) =
xn+1
(−1)
G(x − s) ds.
n
n
(14)
xn
This representation proves the theorem. ✷ In the case where the number of zeros x0 , x1 , . . . , xm is finite, the solution y of Eq. (5) can be represented in the form y(x) = −1 +
m (−1)n Φ(xn − x),
(15)
n=0
∞ G(s) ds x
= e−αx cos βx + (4γ − 1)−1/2 sin βx ≡ Ce−αx cos(βx − δ)
equation tan δ = (4γ − 1)−1/2 . For negative values of x, we have Φ(−x) = 2 − Φ(x).
3. Examples Consider the solution of Eq. (5) that has one zero x0 . For the sake of definiteness, assume that x0 = 0. By (15), the solution has the form y(x) = sign x
× 1 − e−α|x| cos βx + (4γ − 1)−1/2 sin β|x| . (17) It is a kink solution [7,15,16] of Eq. (1) with the piecewise cubic-like nonlinearity (4). Comparing it with the kink solutions of Eq. (2), which have the form √ u = ± tanh(x/ 2 ), solution (17) has a sinuous shape. Consider now a solution that has two zeros. Let y(x) be a solution of (5) given by (15) that has zeros at x = x0 = 0 and x = x1 = a. The solution in this case is a sinuous soliton [7,15,16], y(x) = 1 − Φ(x) − Φ(a − x).
(18)
The condition that solution (18) has zeros at x = 0 and x = a leads to a condition that needs to be imposed on the number a, Φ(a) = 0.
(19)
Solutions of (19), with the use of (16), are found in the form π βa = δ − + nπ, n = 1, 2, . . . , (20) 2 and are uniquely defined by the integer n, βa + 1, n= (21) π where [ξ ] denotes the integer part of the number ξ .
where Φ(x) = 2
301
(16)
for positive values of the argument x, C = (cos δ)−1 . The angle δ (0 < δ < π/2) is determined from the
Conclusion 1. A simple soliton solution of Eq. (5) (a bounded solution with two zeros) is uniquely defined, up to the sign and translation in x, by the integer n, given in (21), where a is the distance between the zeros of the solution and is given by formula (20). Consider also a solution of (5) given by (15), that has three zeros x0 , x1 , x2 . The distances a1 = x1 − x0
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and a2 = x2 − x1 between the zeros satisfy the following system:
These integers n1 and n2 can assume any integer positive values.
Φ(a1 ) = Φ(a1 + a2 ), Φ(a2 ) = Φ(a1 + a2 ).
(22)
It follows from this system that there exists a real constant h such that the equation Φ(x) = h has at least two distinct solutions, x = a1 and x = a1 + a2 . In virtue of (16), this leads to the following a priori estimate for solutions of system (22): βak >
1 π +δ− √ e−λπ ≡ αmin , 2 1 + λ2
k = 1, 2.
(23) By using (16) and estimate (23), we get from system (22) that | cos(βak − δ)| < e−λαmin . This gives the following representation for the solutions a1 , a2 of system (22): π βak = − + δ + nk π + k , (24) 2 where nk are natural numbers and |k | < 1/4. Let us now show that for any natural n1 and n2 , system (22) has a solution admitting representation (24) and the solution is unique. To do that, let us substitute (24) into system (22) and use (16). We get the following system for 1 , 2 , which is equivalent to (22):
1 = arcsin (−1)n2 −1 e−λ(−π/2+δ+n2 π+2 ) × cos(1 + 2 + δ) ,
2 = arcsin (−1)n1 −1 e−λ(−π/2+δ+n1 π+1 ) × cos(1 + 2 + δ) . (25) System (25) can be written as = F (),
(26)
where = (1 , 2 ) is a two-dimensional vector and the function F , which is defined by the right-hand sides of (25), is continuous, maps the square {: |k | π/4} into itself, and is a contraction. This yields existence and uniqueness of solutions of system (25) for any natural n1 and n2 . Conclusion 2. A bounded solution of (5) with three zeros is uniquely, up to the sign and translation in x, defined by two integers n1 and n2 that, together with the distances a1 and a2 , given by (24), satisfy (22).
4. General solution Definition 1. Let y(x; {xk }) be a bounded solution of (5) with zeros {xk } increasingly ordered. A sequence of natural numbers nk = [(β/π)(xk − xk−1 )] + 1 is called a skeleton of the solution y(x; {xk }). Theorem 2. Let y(x, {xk }) be a bounded solution of (5) with a finite number of zeros {xk }m k=0 . Let the distances an = xn − xn−1 between neighboring zeros satisfy an π/β. Then this solution is uniquely defined, up to the sign and translation in x, by its skeleton {nk }m k=1 . An arbitrary finite sequence of natural numbers {nk }m k=1 , nk 2 is a skeleton of a bounded solution of Eq. (5). Proof. By Theorem 1, a solution is uniquely, up to the sign, determined by its zeros. By using the explicit expression for solution (15), one obtains for the distances between zeros, an = xn − xn−1 , the following system of equations: (−1)i+j +1 Φ(ai + · · · + aj ), Φ(ak ) = i
k = 1, . . . , m.
(27)
Using (16), which gives an explicit form of the function Φ, and the a priori estimate βan π which is assumed to hold, we get from system (27) that cos(βak − δ) e−λπ(j −i)
<
i
(n + 1)pn
n=1
= p(2 − p)(1 − p)−2 , where p = e−λπ . This gives a representation of form (24) for solutions of system (27), where the integers nk 2 and |k | < 0.1 (k = 1, . . . , m). Let us now show that, for arbitrary natural nk 2, system (27) has a solution admitting representation (24) with |k | < 0.1 and this solution is unique. By using (24) and (16), system (27) can be transformed to the equivalent system for the vector = (1 , . . . , m )
S. Albeverio, I.L. Nizhnik / Physics Letters A 288 (2001) 299–304
of form (26). This equation has a unique solution, since the function F is continuous, maps the mdimensional cube {: |k | 0.1} into itself, and is a contraction. Indeed, the inequality (2) F − F (1) q (2) − (1) , where we take = max1km |k |, holds for a finite positive constant q which is the product of the Lipschitz constant for the function arcsin x, |x| 0.1, and the sum of absolute values of partial derivatives of the right-hand sides of system (27) with respect to the variables 1 , . . . , m . Hence, we get the estimate q 1.01
∞
(λn + n + 1)(n + 1)p
n
n=1
= 1.01p 2λ + 4 − 3p + p2 (1 − p)−3 < 1, since p = e−λπ and λ 1.
✷
Since the integers {nk }, which by Theorem 2 describe the distances between the neighboring zeros of the bounded solution of Eq. (5), can take any given values, this leads to solutions with “chaotic” behavior in x. Definition 2. We say that Eq. (1) with a bistable nonlinearity f gives rise to a spatial chaos if for any > 0 there exists L() > 0 such that for any sequence of points {(xn , σn )}N n=1 with xn+1 − xn > L() and σn taking one of the two stable values for every n (σn = ±1), Eq. (1) has a bounded stationary solution y such that y(xn ) − σn < , n = 1, . . . , N. (28)
Theorem 2, this can be done. Because the solution y does not change sign between its two zeros, we have that sign y(xm ) = σn , since σm does not change if the points (xm , σm ) are such that yk < xm < yk+1 . The needed estimate (28) for y now follows from inequality (12), since, by the construction, the distance between xn and I and, consequently, all zeros of the solution y exceeds (1/β) ln(4/). ✷
5. Spatial entropy Definition 3. Let S(L) be the number of all bounded solutions of Eq. (5) such that their zeros belong to the interval [0, L] and that the point x = 0 is a zero of all solutions. The number 1 (29) ln S(L), L→∞ L if it exists, is called spatial entropy for Eq. (5) with respect to solutions with a finite number of zeros.
η = lim
Theorem 4. The spatial entropy η for Eq. (5) admits the following estimate: 1 for γ > , 4 where β is given by (8). η > 0.14β > 0,
Proof. Let a sufficiently small > 0 be given. Set L() = (2/β) ln(4/) + 4π/β. Let {(xn , σn )}N n=1 be a sequence from Definition 2. Consider the midpoints yk = (1/2)(xk + xk+1 ) of all the intervals (xk , xk+1 ) such that σk = σk+1 . Consider new intervals Ik with centers in the points yk and length 4π/β. Let I denote the union of the intervals. For Eq. (5), let us construct a bounded solution that has, in I , a finite number of zeros one and only one of which lies in every Ik . By
(30)
Proof. Consider all bounded solutions of Eq. (5) having a finite number of zeros and whose skeletons {nk }m k=1 satisfy the condition that nk 2 and m k=1
Theorem 3. The extended Fisher–Kolmogorov equation (1) with bistable piecewise linear cubic-like nonlinearity (4), for γ > 1/4, gives rise to a spatial chaos.
303
nk 0.95
β L. π
(31)
Since βak < −π/2 + δ + nk π + 0.1 and (31) implies that 2m 0.95(β/π)L, we see that m m π ak nk + 0.1m L. xm − x0 = β k=1
k=1
Hence all zeros of the solutions under consideration lie in an interval of length L. Denote by s(L) the number of distinct ordered sequences of natural numbers (n1 , . . . , nm ), satisfying the condition nk 2 and estimate (31). Since s(L) depends only on the integer part of the number N from the right-hand side of (31) (N = [0.95Lβ/π]), we see
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that s(L) = σ (N), where the function σ (N) satisfies the recurrence relation σ (N) = σ (N − 1) + σ (N − 2) + 1. Hence limN→∞ (1/N) ln σ (N) = ln ζ , where ζ is the 2 greatest √ root of the equation ζ = ζ + 1, i.e., ζ = (1 + 5 )/2. This gives the estimate η 0.95(β/π) ln ζ > 0.14β, since according to Theorem 2 the number s(L) does not exceed the number S(L) from Definition 3 of the spatial entropy. ✷ Remark 4. The established properties of stationary solutions of Eq. (5) and the positive values of the lower estimate (30) for the spatial entropy signify by [3,5,15] the existence of spatial chaos in system (1), (4) under consideration.
Acknowledgements The second author is indebted to SFB-256 for financial support and thanks the Institute of Applied Mathematics at the University of Bonn for hospitality.
References [1] D.G. Aronson, H. Weinberger, Adv. Math. 30 (1978) 33. [2] V. Booth, T. Erneux, Physica A 188 (1992) 206. [3] S.-N. Chow, J. Mallet-Paret, IEEE Trans. Circuits Syst. 42 (10) (1995) 746. [4] L.O. Chua, M. Hasler, G. Moschytz, J. Neirynck, IEEE Trans. Circuits Syst. I 42 (1995) 559. [5] P. Coullet, C. Elphick, D. Repaux, Phys. Rev. Lett. 58 (1987) 431. [6] G.T. Dee, W. van Saarloos, Phys. Rev. Lett. 60 (1988) 2641. [7] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1982. [8] T. Erneux, G. Nicolis, Physica D 67 (1993) 237. [9] P.C. Fife, in: S. Levin (Ed.), Lecture Notes in Biomathematics, Springer, Berlin, 1979. [10] L. Glass, P. Hunter, McCulloch, Theory of Heart, Springer, Berlin, 1991. [11] J.P. Keener, SIAM J. Appl. Math. 47 (3) (1987) 556. [12] A. Kolmogorov, I. Petrovskii, N. Piskunov, Bull. Univ. Moskou Ser. Int. Sec. A 1 (1937) 1. [13] T. Kohonen, Self-Organizing Maps, Springer, Berlin, 1995. [14] V.I. Nekorkin, V.B. Kazantsev, N.F. Rulkov, M.G. Velarde, L.O. Chua, IEEE Trans. Circuits Syst. 42 (10) (1995) 785. [15] L.P. Nizhnik, M. Hasler, I.L. Nizhnik, Preprint 99.4, Kyiv, 1999, Int. J. Bifurc. Chaos, accepted. [16] L.A. Peletier, W.C. Troy, CWI Quart. 9 (1996) 121. [17] J.A. Powell, A. Newell, C.K.R.T. Jones, Phys. Rev. A 44 (1991) 3636.
Physics Letters A 288 (2001) 299–304 www.elsevier.com/locate/pla
Spatial chaos in a fourth-order nonlinear parabolic equation S. Albeverio a , I.L. Nizhnik b,∗ a Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53 115 Bonn 1, Germany b Institute of Mathematics, 3 Tereschenkivska Str., Kiev 01 601, Ukraine
Received 14 March 2001; received in revised form 26 June 2001; accepted 8 August 2001 Communicated by A.P. Fordy
Abstract Bounded stationary solutions for a fourth-order extended Fisher–Kolmogorov equation with cubic-like piecewise linear nonlinearity are given. Those solutions which have a finite number of zeros are characterized by means of a set of integers associated with the distance between the zeros. The value of the spatial entropy is estimated and the existence of spatial chaos is shown. 2001 Elsevier Science B.V. All rights reserved. AMS classification: 35C05; 35G25; 35K57; 35K60; 58F13 Keywords: Diffusion equation; Entropy; Spatial chaos
1. Introduction In this Letter, we will be concerned with stationary solutions of bistable systems described by the extended Fisher–Kolmogorov equation ∂u ∂ 4u ∂ 2u (1) = −γ 4 + 2 − f (u), γ > 0, ∂t ∂x ∂x where the function f will be specified below as a cubic-like piecewise linear nonlinear one. Eq. (1) where f (u) = u3 − u was proposed [5,6] as a generalization of the classical Fisher–Kolmogorov equation ∂u ∂ 2 u = 2 + u − u3 . (2) ∂t ∂x The latter equation has a bistable nonlinearity in the sense that the ordinary differential equation du/dt = u − u3 has two stable stationary solutions, u = ±1. * Corresponding author.
E-mail address: [email protected] (I.L. Nizhnik).
It plays an important role in various areas of physics, chemistry, and biology [1,12,17]. The presence of a bistable nonlinearity in the diffusion equation is the cause for the appearance of various interesting phenomena such as self-organization, pattern formation and spatial chaos, see, e.g., [2–4,8–11, 13] and references therein. It was shown in [11,14,15] that for a difference analogue of Eq. (2), ∂un = d[un+1 + un−1 − 2un ] + un − u3n , (3) ∂t with a small value of the cell coupling constant d 0, stable stationary solutions un are close to the signature function σn that takes values +1 or −1 for each n such that |un − σn | < 4d. Here, every stable stationary solution un is uniquely determined by its skeleton, uˆ n = sign un , and an arbitrary signature function is the skeleton of a stationary stable solution. This result extends to the case of
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 5 3 2 - 1
300
S. Albeverio, I.L. Nizhnik / Physics Letters A 288 (2001) 299–304
Eq. (3) with an arbitrary value of d and the piecewise linear cubic-like nonlinearity [15]: u − 1, u > 0, f (u) = u + 1, u < 0, (4) 0, u = 0. Nonlinearity (4) is also a bistable nonlinearity. It reproduces features of a cubic nonlinearity, and allows to obtain exact results in terms of explicit formulas. In particular, an estimate for the spatial entropy was obtained for Eq. (3) with nonlinearity (4). It has also been proved that there is a spatial chaos in system (3) [15]. For Eq. (2), which is a continuous analogue of Eq. (3), all bounded stationary solutions can be explicitly expressed in terms of elliptic functions and are reduced to periodic solutions or kinks. Thus bounded stationary solutions of Eq. (2) do not have chaotic behavior, as opposed to solutions of the discrete analogue equation (3). In this Letter, we prove that there is an effect of spatial chaos appearing in solutions of the fourthorder equation (1) with the piecewise linear cubic-like nonlinearity (4). Our method is based on the study of the location of zeros of stationary solutions of (1). This constructive approach is fundamentally different from the analysis of the qualitative behavior of solutions of the stationary equation (1) with a cubic nonlinearity which is given in [16].
We will study bounded stationary solutions of the extended Fisher–Kolmogorov equation (1) with the piecewise linear nonlinearity (4): 1 γ> . (5) 4 By definition we set sign 0 = 0. If y is a solution of Eq. (5), then −y is also a solution of this equation. Hence, we will be looking for a solution up to the sign. The linear equation corresponding to (5), γ y (4) − y
+ y = sign y,
γ y (4) − y
+ y = 0,
(6)
has four linearly independent solutions:
eαx sin βx.
e−αx sin βx,
+∞ y(x) = G(x − s) sign y(s) ds,
(9)
−∞
where G is explicitly expressed via solution (7) of the homogeneous equation (6): 1 G(x) = √ e−α|x| cos βx + λ sin β|x| . (10) 4α γ Remark 1. From representation (9) and expression (10) one has the a priori estimate for the bounded solution of Eq. (5), y(x)
+∞ G(s) ds
−∞
√ 4 γ √ , 1+2 γ
because of | cos βx| 1 and | sin βx| β|x|. By using representation (9), it is easy to get the following lemma.
2. Stationary solutions
e−αx cos βx,
The numbers α > 0 and β > 0 are found in terms of the parameter γ from (6) by 1 1 1/2 1 1/2 1 + − , β = , α= √ √ 2 γ 4γ 2 γ 4γ α λ = > 1. (8) β Bounded solutions of Eq. (5), for a given function sign y, can be represented in terms of the Green’s function G in the form of well-defined Lebesgue integral:
eαx cos βx, (7)
Lemma 1. Let y ≡ 0 be a bounded solution of Eq. (5) not changing sign on an interval [x0 − l, x0 + l]. Then, for all x ∈ [x0 − l, x0 + l] save for, possibly, a finite number of points, the function sign y(x) takes a constant value σ = ±1. We also have the following estimate: ∞ y(x0) − σ 4 G(s) ds < 2(1 + λ)e−λβl . (11) l
Remark 2. Let y be a bounded solution of Eq. (5) and let the distance l between a point x and the closest zero of the solution y satisfy the inequality βl ln(4/), 0 < 1. Then y(x) − sign y(x) < . (12)
S. Albeverio, I.L. Nizhnik / Physics Letters A 288 (2001) 299–304
Remark 3. For βl > ln 4, it follows from inequality (11) that |y(x0 ) − σ | < 1 and, hence, y(x0 ) = 0. Thus, if the distance l between neighboring zeros of a bounded solution y of Eq. (5) satisfies l > (1/β) ln 4, then the solution changes sign in a neighborhood of each zero. Theorem 1. Let {xk } be the increasingly ordered set of zeros of a bounded solution y = y(x, {xk }) of Eq. (5). Let a solution change its sign in a neighborhood of each zero xn . Then this solution is uniquely defined (up to the sign) by its zeros. Proof. By using the zeros {xk } of the solution y, it is possible to uniquely (up to the sign) construct the function sign y(x), without loss of generality assuming that 1, x2n < x < x2n+1 , sign y(x) = (13) −1, x2n+1 < x < x2n+2 , otherwise instead of y(x) we take −y(x). If the set of the zeros {xn } in (13) is bounded from below (above), it will be augmented by −∞ (+∞). Substitution of (13) into (9) gives an explicit expression for the solution y of Eq. (5) in terms of its zeros {xk } in the form of the absolutely convergent series: y(x) =
xn+1
(−1)
G(x − s) ds.
n
n
(14)
xn
This representation proves the theorem. ✷ In the case where the number of zeros x0 , x1 , . . . , xm is finite, the solution y of Eq. (5) can be represented in the form y(x) = −1 +
m (−1)n Φ(xn − x),
(15)
n=0
∞ G(s) ds x
= e−αx cos βx + (4γ − 1)−1/2 sin βx ≡ Ce−αx cos(βx − δ)
equation tan δ = (4γ − 1)−1/2 . For negative values of x, we have Φ(−x) = 2 − Φ(x).
3. Examples Consider the solution of Eq. (5) that has one zero x0 . For the sake of definiteness, assume that x0 = 0. By (15), the solution has the form y(x) = sign x
× 1 − e−α|x| cos βx + (4γ − 1)−1/2 sin β|x| . (17) It is a kink solution [7,15,16] of Eq. (1) with the piecewise cubic-like nonlinearity (4). Comparing it with the kink solutions of Eq. (2), which have the form √ u = ± tanh(x/ 2 ), solution (17) has a sinuous shape. Consider now a solution that has two zeros. Let y(x) be a solution of (5) given by (15) that has zeros at x = x0 = 0 and x = x1 = a. The solution in this case is a sinuous soliton [7,15,16], y(x) = 1 − Φ(x) − Φ(a − x).
(18)
The condition that solution (18) has zeros at x = 0 and x = a leads to a condition that needs to be imposed on the number a, Φ(a) = 0.
(19)
Solutions of (19), with the use of (16), are found in the form π βa = δ − + nπ, n = 1, 2, . . . , (20) 2 and are uniquely defined by the integer n, βa + 1, n= (21) π where [ξ ] denotes the integer part of the number ξ .
where Φ(x) = 2
301
(16)
for positive values of the argument x, C = (cos δ)−1 . The angle δ (0 < δ < π/2) is determined from the
Conclusion 1. A simple soliton solution of Eq. (5) (a bounded solution with two zeros) is uniquely defined, up to the sign and translation in x, by the integer n, given in (21), where a is the distance between the zeros of the solution and is given by formula (20). Consider also a solution of (5) given by (15), that has three zeros x0 , x1 , x2 . The distances a1 = x1 − x0
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and a2 = x2 − x1 between the zeros satisfy the following system:
These integers n1 and n2 can assume any integer positive values.
Φ(a1 ) = Φ(a1 + a2 ), Φ(a2 ) = Φ(a1 + a2 ).
(22)
It follows from this system that there exists a real constant h such that the equation Φ(x) = h has at least two distinct solutions, x = a1 and x = a1 + a2 . In virtue of (16), this leads to the following a priori estimate for solutions of system (22): βak >
1 π +δ− √ e−λπ ≡ αmin , 2 1 + λ2
k = 1, 2.
(23) By using (16) and estimate (23), we get from system (22) that | cos(βak − δ)| < e−λαmin . This gives the following representation for the solutions a1 , a2 of system (22): π βak = − + δ + nk π + k , (24) 2 where nk are natural numbers and |k | < 1/4. Let us now show that for any natural n1 and n2 , system (22) has a solution admitting representation (24) and the solution is unique. To do that, let us substitute (24) into system (22) and use (16). We get the following system for 1 , 2 , which is equivalent to (22):
1 = arcsin (−1)n2 −1 e−λ(−π/2+δ+n2 π+2 ) × cos(1 + 2 + δ) ,
2 = arcsin (−1)n1 −1 e−λ(−π/2+δ+n1 π+1 ) × cos(1 + 2 + δ) . (25) System (25) can be written as = F (),
(26)
where = (1 , 2 ) is a two-dimensional vector and the function F , which is defined by the right-hand sides of (25), is continuous, maps the square {: |k | π/4} into itself, and is a contraction. This yields existence and uniqueness of solutions of system (25) for any natural n1 and n2 . Conclusion 2. A bounded solution of (5) with three zeros is uniquely, up to the sign and translation in x, defined by two integers n1 and n2 that, together with the distances a1 and a2 , given by (24), satisfy (22).
4. General solution Definition 1. Let y(x; {xk }) be a bounded solution of (5) with zeros {xk } increasingly ordered. A sequence of natural numbers nk = [(β/π)(xk − xk−1 )] + 1 is called a skeleton of the solution y(x; {xk }). Theorem 2. Let y(x, {xk }) be a bounded solution of (5) with a finite number of zeros {xk }m k=0 . Let the distances an = xn − xn−1 between neighboring zeros satisfy an π/β. Then this solution is uniquely defined, up to the sign and translation in x, by its skeleton {nk }m k=1 . An arbitrary finite sequence of natural numbers {nk }m k=1 , nk 2 is a skeleton of a bounded solution of Eq. (5). Proof. By Theorem 1, a solution is uniquely, up to the sign, determined by its zeros. By using the explicit expression for solution (15), one obtains for the distances between zeros, an = xn − xn−1 , the following system of equations: (−1)i+j +1 Φ(ai + · · · + aj ), Φ(ak ) = i
k = 1, . . . , m.
(27)
Using (16), which gives an explicit form of the function Φ, and the a priori estimate βan π which is assumed to hold, we get from system (27) that cos(βak − δ) e−λπ(j −i)
<
i
(n + 1)pn
n=1
= p(2 − p)(1 − p)−2 , where p = e−λπ . This gives a representation of form (24) for solutions of system (27), where the integers nk 2 and |k | < 0.1 (k = 1, . . . , m). Let us now show that, for arbitrary natural nk 2, system (27) has a solution admitting representation (24) with |k | < 0.1 and this solution is unique. By using (24) and (16), system (27) can be transformed to the equivalent system for the vector = (1 , . . . , m )
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of form (26). This equation has a unique solution, since the function F is continuous, maps the mdimensional cube {: |k | 0.1} into itself, and is a contraction. Indeed, the inequality (2) F − F (1) q (2) − (1) , where we take = max1km |k |, holds for a finite positive constant q which is the product of the Lipschitz constant for the function arcsin x, |x| 0.1, and the sum of absolute values of partial derivatives of the right-hand sides of system (27) with respect to the variables 1 , . . . , m . Hence, we get the estimate q 1.01
∞
(λn + n + 1)(n + 1)p
n
n=1
= 1.01p 2λ + 4 − 3p + p2 (1 − p)−3 < 1, since p = e−λπ and λ 1.
✷
Since the integers {nk }, which by Theorem 2 describe the distances between the neighboring zeros of the bounded solution of Eq. (5), can take any given values, this leads to solutions with “chaotic” behavior in x. Definition 2. We say that Eq. (1) with a bistable nonlinearity f gives rise to a spatial chaos if for any > 0 there exists L() > 0 such that for any sequence of points {(xn , σn )}N n=1 with xn+1 − xn > L() and σn taking one of the two stable values for every n (σn = ±1), Eq. (1) has a bounded stationary solution y such that y(xn ) − σn < , n = 1, . . . , N. (28)
Theorem 2, this can be done. Because the solution y does not change sign between its two zeros, we have that sign y(xm ) = σn , since σm does not change if the points (xm , σm ) are such that yk < xm < yk+1 . The needed estimate (28) for y now follows from inequality (12), since, by the construction, the distance between xn and I and, consequently, all zeros of the solution y exceeds (1/β) ln(4/). ✷
5. Spatial entropy Definition 3. Let S(L) be the number of all bounded solutions of Eq. (5) such that their zeros belong to the interval [0, L] and that the point x = 0 is a zero of all solutions. The number 1 (29) ln S(L), L→∞ L if it exists, is called spatial entropy for Eq. (5) with respect to solutions with a finite number of zeros.
η = lim
Theorem 4. The spatial entropy η for Eq. (5) admits the following estimate: 1 for γ > , 4 where β is given by (8). η > 0.14β > 0,
Proof. Let a sufficiently small > 0 be given. Set L() = (2/β) ln(4/) + 4π/β. Let {(xn , σn )}N n=1 be a sequence from Definition 2. Consider the midpoints yk = (1/2)(xk + xk+1 ) of all the intervals (xk , xk+1 ) such that σk = σk+1 . Consider new intervals Ik with centers in the points yk and length 4π/β. Let I denote the union of the intervals. For Eq. (5), let us construct a bounded solution that has, in I , a finite number of zeros one and only one of which lies in every Ik . By
(30)
Proof. Consider all bounded solutions of Eq. (5) having a finite number of zeros and whose skeletons {nk }m k=1 satisfy the condition that nk 2 and m k=1
Theorem 3. The extended Fisher–Kolmogorov equation (1) with bistable piecewise linear cubic-like nonlinearity (4), for γ > 1/4, gives rise to a spatial chaos.
303
nk 0.95
β L. π
(31)
Since βak < −π/2 + δ + nk π + 0.1 and (31) implies that 2m 0.95(β/π)L, we see that m m π ak nk + 0.1m L. xm − x0 = β k=1
k=1
Hence all zeros of the solutions under consideration lie in an interval of length L. Denote by s(L) the number of distinct ordered sequences of natural numbers (n1 , . . . , nm ), satisfying the condition nk 2 and estimate (31). Since s(L) depends only on the integer part of the number N from the right-hand side of (31) (N = [0.95Lβ/π]), we see
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that s(L) = σ (N), where the function σ (N) satisfies the recurrence relation σ (N) = σ (N − 1) + σ (N − 2) + 1. Hence limN→∞ (1/N) ln σ (N) = ln ζ , where ζ is the 2 greatest √ root of the equation ζ = ζ + 1, i.e., ζ = (1 + 5 )/2. This gives the estimate η 0.95(β/π) ln ζ > 0.14β, since according to Theorem 2 the number s(L) does not exceed the number S(L) from Definition 3 of the spatial entropy. ✷ Remark 4. The established properties of stationary solutions of Eq. (5) and the positive values of the lower estimate (30) for the spatial entropy signify by [3,5,15] the existence of spatial chaos in system (1), (4) under consideration.
Acknowledgements The second author is indebted to SFB-256 for financial support and thanks the Institute of Applied Mathematics at the University of Bonn for hospitality.
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