- Email: [email protected]

Persistence of travelling waves for a coupled nonlinear wave system Junliang Lu a

a,b,*

, Tianlan He a, Dahe Feng

a

Center of Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, China b Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL, 35899, USA

Abstract In this paper, we study a sort of coupled nonlinear wave system. Special attention is paid to the question of the existence of heteroclinic orbits of the associated ordinary diﬀerential equation from the geometric singular perturbation point of view. We prove that a solitary wave persists when the perturbation parameter is suitably small. This argument does not require an explicit expression for the original coupled nonlinear wave system. 2007 Published by Elsevier Inc. Keywords: Coupled nonlinear wave system; Travelling waves; Geometric singular perturbation theory; Nonlinear wave equations; Equilibrium; Normally hyperbolic manifold

1. Introduction Geometric singular perturbation theory arises in a variety of applications in many ﬁelds such as two-point boundary value problems, travelling wave problems in reaction–diﬀusion equations, chemical pattern formation, the propagation of action potentials in neurophysiology, coupled mechanical oscillators, perturbed Hamiltonian systems, adiabatic Hamiltonian systems, combustion friction modelling, optics, ﬂuid particle motion in Lagrangian framework for ﬂuid mechanics, ray theory for wave propagation,celestial mechanics, capture into resonance, control theory, material science, etc. For nonlinear wave equations, a great deal of investigation on explicit and exact solitary wave solutions has been made in the last three decades. One can easily ﬁnd abundant methods and reports about nonlinear equations. However, when talking about perturbed wave equations, especially singularly perturbed ones, the ﬁrst question is about the existence and persistence of solitary waves. Compared with traditional methods, geometric singular perturbation methods play a special role in giving a ﬁrst picture of the perturbed solutions. Geometric singular perturbation theory was ﬁrst given by Fenichel [1], and is often referred to as Fenichel Theory. This theory exploits a diﬀerential equation’s

* Corresponding author. Address: Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL, 35899, USA. E-mail address: [email protected] (J. Lu).

0096-3003/$ - see front matter 2007 Published by Elsevier Inc. doi:10.1016/j.amc.2007.02.092

348

J. Lu et al. / Applied Mathematics and Computation 191 (2007) 347–352

geometric structures, such as its slow (or center) manifolds and its fast stable and unstable ﬁbers. Fenichel’s geometric theory has already been successfully applied in some nonlinear systems. Jones [2] used it to prove the existence of the homoclinic or heteroclinic orbits corresponding to travelling wave solutions in a nonlinear reaction–diﬀusion equation; Guo et al. [3] studied the nonlinear Schro¨dinger equation via Fenichel’s method; Fan et al. [4] studied the existence of solitary waves of singularly perturbed mKdV–KS equation via Fenichel’s method; Kyrychko et al. [5] studied the persistence of travelling wave solutions of a fourth order diﬀusion system via geometric singular perturbation theory. In this paper, we shall use Fenichel persistence theory to study the question of existence of solutions for damped coupled nonlinear wave equations. In 2002, Fang and Guo [6] consider the following time periodic problem of damped nonlinear wave equations: ut þ f ðuÞx auxx þ buxxx þ 2vvx ¼ G1 ðu; vÞ þ h1 ðxÞ; ð1Þ vt cvxx þ 2ðuvÞx þ gðvÞx ¼ G2 ðu; vÞ þ h2 ðxÞ; where a, b and c are constant,and c > 0; b 6¼ 0. Under the periodic boundary conditions, the authors obtained the unique existence of strong solutions for the above system. In the paper of Zhang and Li [7], the authors investigated the above system (1) with Gi ðu; vÞ 0; hi ðxÞ 0; i ¼ 1; 2, and obtained bifurcations of travelling wave solutions of this system. For system (1), let n ¼ x ct;

u ¼ uðx ctÞ;

v ¼ vðx ctÞ;

where c > 0 is the wave speed. Substituting the above travelling solutions into (1), we have cu0 þ f 0 ðuÞ au00 þ bu000 þ 2vv0 ¼ 0; cv0 þ 2ðuvÞ0 cv00 þ g0 ðvÞ ¼ 0;

ð2Þ

ð3Þ

where the prime denotes diﬀerentiation with respect to n. Integrating (2) with respect to n once, we obtain cu þ f ðuÞ au0 þ bu00 þ v2 ¼ 0; ð4Þ cv þ 2uv cv0 þ gðvÞ ¼ 0: Write that x1 ¼ u; x2 ¼ u0 ; and x3 ¼ v, Eq. (3) is equivalent to the following three-dimensional system 8 dx 1 ¼ x2 ; > > < dn dx2 b dn ¼ cx1 þ ax2 x23 f ðx1 Þ; > > : dx3 cx3 þ2x1 x2 þgðx3 Þ ¼ ; dn c

ð5Þ

where c > 0; b 6¼ 0. When 0 < b 1, (5) is a singularly perturbed equation. This paper is organized as follows: In Section 2, we make a brief introduction to Fenichel’s geometric singular perturbation theory because it seems powerful to deal with singularly perturbed problems. In Section 3, we analyze the equilibria of damped coupled nonlinear wave equations. And in Section 4, using geometric singular perturbation theory, we point out the persistence of the heteroclinic orbits. 2. Fenichel theory In this section, we review the necessary theory that we will use for our discussion. We take the exposition in Kaper [8], for detail, one can resort to Fenichel [1]. Consider the standard fast-slow system: 0 x ¼ fðx; z; Þ; ð6Þ z0 ¼ gðx; z; Þ: Here, x 2 Rm ; z 2 Rn ; 2 R; and the prime denotes diﬀerentiation with respect to the independent variable t. The functions f and g are assumed to be C1 function of x, z and in U · I, where U is an open subset of Rm Rn and I is an open interval containing = 0.

J. Lu et al. / Applied Mathematics and Computation 191 (2007) 347–352

349

It is useful at various stages to study a reformulation of the system (6) in terms of the rescaled variable s ¼ t: x_ ¼ fðx; z; Þ; ð7Þ z_ ¼ gðx; z; Þ; where the dot denotes diﬀerentiation with respect to the new independent variable s. As long as 6¼ 0, systems (6) and (7) are equivalent. The independent variable t and s are referred to as the fast and slowtimes, respectively; (6) and (7) are called the fast and slowsystems, respectively; x is called a fast variable, and z is called a slow variable. We make the following assumptions about (6): (A1) The function f and g are C1 in U · I, where U is an open subset of Rm Rn and I is an open interval containing = 0. (A2) There exists a set M0 that is contained in fðx; zÞ : fðx; z:0 ¼ 0Þg such that M0 is a compact manifold with boundary and M0 is given by the graph of a C1 function x ¼ X0 ðzÞ for z 2 D, where D Rn is a compact, simply connected domain and the boundary of D is an (n 1)-dimensional C1 submanifold. Finally, the set D is overﬂowing invariant with respect to (7) when = 0. (A3) M0 is normally hyperbolic relative to (6) when = 0. Deﬁnition (Fenichel). A manifold M is said to be normally hyperbolic for a system of diﬀerential equations if the linearization of those diﬀerential equations at each point on M has exactly c eigenvalues with zero real part, where c is the dimension of the center dimensions. Theorem 1 (Fenichel’s Persistence Theorem). Let system (6) satisfy the conditions (A1)–(A3). If > 0 is sufficiently small, then there exists a function Xðz; Þ defined on D such that the manifold M ¼ fðx; zÞ : x ¼ Xðz; Þg is locally invariant under (6). Moreover, Xðz; Þ is Cr for any r < þ1, and M is C r OðÞ close to M0. In addition, there exist perturbed local stable and unstable manifolds of M. They are the unions of invariant families of fast stable and unstable fibers of dimension l and k, respectively, and they are C r OðÞ close, for all r < þ1, to their unperturbed counterparts. 3. Equilibria analysis For system (5), we consider f ðx1 Þ ¼ x21 ; gðx3 Þ ¼ x23 , then (5) can be transformed into 8 dx1 ¼ x2 ; > dn > > < 2 ¼ cx1 þ ax2 x23 x21 ; b dx dn > > > : dx3 cx3 þ2x1 x2 þx23 ¼ ; dn c Let n ¼ cs. Then system (8) can be transformed into 8 dx 1 ¼ bx2 ; > > ds > < dx2 ¼ cx1 þ ax2 x23 x21 ; ds > > > dx3 cx þ2x x þx2 : ¼ b 1 c1 2 3 ; ds System (9) has equilibria: E0 ¼ ð0; 0; 0Þ; E1 ¼ ðc; 0; 0Þ. At E0, the eigenvalues are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ bc a a2 þ 4bc 0 0 ; k1 ¼ ; k2;3 ¼ c 2

ð8Þ

ð9Þ

350

J. Lu et al. / Applied Mathematics and Computation 191 (2007) 347–352

Likewise, at E1, the eigenvalues are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ bc a a2 4bc 1 1 k1 ¼ ; k2;3 ¼ ; c 2 when a 6¼ 0; c 6¼ 0; E0 ; E1 are hyperbolic equilibria. Below, we shall prove the persistence of a travelling wave solution which passes through E0 ¼ ð0; 0; 0Þ and E1 ¼ ðc; 0; 0Þ. 4. The ﬂow on the manifold Mb For b = 0, 0 0 0 B @c a 0 0

(9) has Jacobian at E0 1 0 C 0A 0

which has two zero eigenvalues for any c 6¼ 0 or a 6¼ 0. So M0 is normally hyperbolic. Fenichel’s theory then guarantees that there exists a submanifold of R3, Mb that lies within OðbÞ of M0 and is diﬀeomorphic to M0. It is invariant under the ﬂow of (9) and Cr smooth for any r < þ1. When b = 0, x2 must lie on the set cx1 þ x21 þ x23 3 M 0 ¼ ðx1 ; x2 ; x3 Þ 2 R : x2 ¼ ; a which is a two-dimensional submanifold in R3. Next, we determine the dynamics on Mb. In order to do it, let us write M b ¼ fðx1 ; x2 ; x3 Þ 2 R3 : x2 ¼ hðx1 ; x3 ; bÞg; where the function h satisﬁes: hðx1 ; x3 ; 0Þ ¼ x21

cx1 þx21 þx23 . a

ð10Þ Now, we expand the Taylor series of h in the variable b:

x23

cx1 þ þ þ bh1 ðx1 ; x3 ; 0Þ þ Oðb2 Þ: a Diﬀerentiating (11) and substituting (8), we obtain the following system: oh ox1 oh ox3 þ b ¼ cx1 þ ah x23 x21 : ox1 on ox3 on hðx1 ; x3 ; bÞ ¼

ð11Þ

Powers of b give h1 ¼

c2 cx1 ð3cc þ 2caÞx21 ccx23 þ 2cx31 þ 2cx1 x23 þ 4cx21 x3 þ 4x31 x3 þ 4x1 x33 : a2 c

And the system (8) becomes 8 2 2 < dx1 ¼ cx1 þx1 þx3 þ bhðx1 ; x3 ; 0Þ þ Oðb2 Þ; dn

a

: dx3 ¼ acx3 2cx21 þ2x31 þ2x1 x23 þ 2bx1 h ðx ; x ; 0Þ þ Oðb2 Þ: 1 1 3 dn ac c These equations determine the dynamics on the ‘‘slow’’ manifold Mb. When b = 0, system (12) reduces to the system of coupled ﬁrst-order ODEs 8 2 2 < dx1 ¼ cx1 þx1 þx3 ; dn a : dx3 ¼ acx3 2cx21 þ2x31 þ2x1 x23 ; dn ac which have equilibria (0, 0) and ðc; 0Þ. At (0, 0) and ðc; 0Þ, the Jocabian are ! ! c 0 ac 0 a and ; 2c2 0 cc cc ac respectively.

ð12Þ

ð13Þ

ð14Þ

J. Lu et al. / Applied Mathematics and Computation 191 (2007) 347–352

351

Below, we prove there exists a heteroclinic connection between the critical points (0, 0) and ðc; 0Þ of (14). This connection corresponds to a travelling wave solution of (1). Firstly, set a > 0; c > 0, then, (0, 0) is a stable node and ðc; 0Þ is a saddle point. Consider the inﬁnite equilibria of (14) below. Via the transformation 1 x1 ¼ ; z

u x3 ¼ ; z

then, (14) becomes 8 2 cu3 zacuz2 < du ¼ 2þ2czþ2u2 cuzþcuz ; 2 dn

acz

ð15Þ

: dz ¼ z2 þcz32u2 z2 : dn az Setting n ¼ z2 s, (15) becomes ( 2 2 cu3 zacuz2 du ¼ 2þ2czþ2u cuzþcuz ; ds ac dz ds

¼ z

2 þcz3 u2 z2

a

ð16Þ

:

When z = 0, (16) does not have equilibria. In addition, via the transformation v x1 ¼ ; z

1 x3 ¼ ; z

then, (14) becomes 8 < dv ¼ czv2 þðcaccÞvz2 2 2v4 2cv3 z ; dn

acz

ð17Þ

3 2v3 z : dz ¼ 2vzþ2cv2 zþacz : dn acz2

Setting n ¼ z2 s, (17) becomes 8 < dv ¼ czv2 þðcaccÞvz2 2v4 2cv3 z ; ds ac

ð18Þ

: dz ¼ 2vzþ2cv2 zþacz3 2v3 z ; ds

ac

which has equilibrium Oð0; 0Þ. By the vector theory, the d-neighborhood S d ðOÞ of equilibrium Oð0; 0Þ consists of a hyper sector and a ellipse sector. From the global phase, we can obtain there exists a heteroclinic connection between the critical points ð0; 0Þ and ðc; 0Þ. Setting c ¼ cð0Þ (where c is a function in b), ðx01 ; x03 Þ is the heteroclinic connection. We now employ the Fredholm theory to show that, for b > 0 suﬃciently small, there exists a hereroclinic connection between the equilibria (0, 0) and ðc; 0Þ. To seek such a connection, set x1 ¼ x01 þ b xe1 ;

x3 ¼ x03 þ b xe3

and substitute into (13). To the lowest order in b, the system governing ð xe1 ; xe3 Þ is 0 1 ! 2x03 cþ2x01 h1 ðx01 ; x03 ; 0Þ e x xe1 d 1 a a A @ ¼ 2 0 2 2 4cx01 þ6x01 þ2x03 acþ4x01 x03 x h ðx0 ; x0 ; 0Þ dn xe3 xe3 c 1 1 1 3 ac

ð19Þ

ac

and we want to prove this system has a solution satisfying xe1 ; xe3 ! 0

as n ! 1:

By Fredholm theory, the system (19) has a square-integrable solution if and only if the following compatibility condition holds

352

J. Lu et al. / Applied Mathematics and Computation 191 (2007) 347–352

Z

þ1

XðnÞ; 1

h1 ðx01 ðnÞ; x03 ðnÞ; 0Þ 2 0 x ðnÞh1 ðx01 ðnÞ; x03 ðnÞ; 0Þ c 1

!! dn ¼ 0

for all functions XðnÞ in the kernel of the adjoint of the operator deﬁned by the left-hand side of (19). The adjoint system for (19) has the form 0 2 2 1 4cx01 þ6x01 þ2x03 cþ2x0 dX @ a 1 ac AX: ¼ ð20Þ 2x03 acþ4x01 x03 dn a ac As n ! þ1, we have x01 ! 0; x03 ! 0, and the matrix in (20) is then a constant matrix with eigenvalues k1 ¼ ac ; k2 ¼ cc. Because a > 0; c > 0, and c > 0, we can see that both eigenvalues c1 and c2 are positive, and as n ! þ1 any solution of (20), other than the trivial solution, must grow exponentially. Therefore, the only solution in L2 is the zero solution XðnÞ ¼ 0, and the Fredholm orthogonality condition holds. Thus, we have proven the existence of the desired connection on the manifold Mb. Secondly, when a < 0, the equilibria of (14), (0, 0) is a saddle and ðc; 0Þ is a node. Similar to the proof above, we can show there exists the heteroclinic connection between (0, 0) and ðc; 0Þ. These results are summarized in the following theorem. Theorem 2. For c > 0, there exists b0 such that for every b 2 ð0; b0 , Eq. (1) admits a travelling wave solution uðx; tÞ ¼ uðnÞ; vðx; tÞ ¼ vðnÞ satisfying uðþ1Þ ¼ 0; vðþ1Þ ¼ 0, and uð1Þ ¼ c; vð1Þ ¼ 0; where n ¼ x ct. 5. Conclusions Using a travelling wave transformation, we obtain an ODE from a coupled nonlinear system. Then using vector theory, we obtain a heteroclinic orbit which is a travelling wave solution of the coupled nonlinear system. In many ﬁelds, travelling wave solutions are always important for the nonlinear system, therefore it is natural to ask a question about the persistence of the travelling wave solution in the nonlinear system. Considering Eq. (1), with the help of the invariant manifold theory and geometric singular perturbation theory, we have proven (for b 1) the persistence of the travelling wave solutions. References [1] N. Fenichel, Geometric singular perturbation theory for ordinary diﬀerential equations, J. Diﬀerential Equations 31 (1979) 53–98. [2] C.K.R.T. Jones, Geometric singular perturbation theory, in: Montecatini Terme, L. Arnold (Eds.), Dynamical Systems, Lecture Notes in Mathematica, vol. 1609, Springer-Verlag, Berlin, 1994, pp. 44–118. [3] Boling Guo, H. Chen, Homoclinic orbit in a six-dimensional model of a perturbed nonlinear Schro¨dinger equation, Commun. Nonlinear Sci. Numer. Simul 9 (4) (2004) 431–442. [4] Xinghua Fan, Lixin Xian, The existence of solitary waves of singularly perturbed mKdV–KS equation, Chaos Solitons Fractals 26 (2005) 1111–1118. [5] Y.N. Kyrycko, M.V. Bartuccelli, K.B. Blyuss, Persistence of travelling wave solutions of a fourth order diﬀusion system, J. Comput. Appl. Math. 176 (2005) 433–443. [6] Shaomei Fang, Boling Guo, Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations, Appl. Math. Mech. (English Edition) 24 (6) (2003) 673–683. [7] Jixiang Zhang, Jibin Li, Bifurcations of travelling wave solutions for a coupled nonlinear wave system, Appl. Math. Mech. (English Edition) 26 (7) (2005) 838–847. [8] T.J. Kaper, An introduction the geometric methods and dynamical systems theory for singular perturbation problems, Proc. Sympos. Appl. Math. 56 (1999) 85–131.

Copyright © 2019 KUNDOC.COM. All rights reserved.