EXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 9 EXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION JIBIN LI ABSTRACT. For the cubic-quintic generalized nonlinear Schrödinger equation (GNLS), the exact dark soliton solution and periodic solutions are given. The persistence of dark solition and chaotic dynamics of the traveling wave solutions for the perturbed cubic-quintic GNLS-equation are rigorously proved. To guarantee the existence of the above solutions, all parameter conditions are determined. 1 Introduction Dark solitons have been studied by a lot of mathematicians and physicists (see the review paper [3] and references therein). More recently, Pelinovsky and Kevrekidis [6] considered the persistence and stability of dark solitons in the Gross-Pitaevskii (GP) equation (1) iu t = 1 u xx + f( u )u + ɛv (x)u, which is a perturbed equation of the generalized nonlinear Schrödinger equation (GNLS) () iu t = 1 u xx + f( u )u. This so called a dark soliton or a black soliton, for which the authors of [6] gave the following definitions. This research was supported by the National Natural Science Foundation of China ( ) and (18313). AMS subject classification: () 34C3, 34C37, 35Q55, 37K45, 58Z5, 74J3. Keywords: Perturbed GNLS equation, singular traveling wave equation, dark soliton, chaos wave, exact explicit solution. Copyright c Applied Mathematics Institute, University of Alberta. 161

2 16 JIBIN LI Definition 1. Any traveling wave solution of the NLS equation () in the form (3) u(x, t) = U(x vt)e iωt, U(z) = Φ(z)e iθ(z), z = x vt, is called a dark soliton if U : R C, Φ : R R +, and Θ : R [ π, π] are smooth functions of their arguments, which converge exponentially fast to the boundary conditions (4) lim z ± Φ(z) = q, lim Θ(z) = Θ ±. z ± Here (ω, v) D R, q I R + and Θ ± [ π, π] are parameters of the solution. Moreover, the functions Φ(z) and Θ(z) can be chosen to satisfy the normalization conditions Φ () = and Θ + =. In addition, it requires that Φ(z) < q on z R. Definition. The limit solution φ (x) = lim v Φ(z) in a family of dark solitons of Definition 1 is said to be a black soliton if φ (x) is a real smooth function on x R. The black soliton is called a bubble if φ ( x) = φ (x) with < φ () < q and lim x ± φ (x) = q, while it is called a kink if φ ( x) = φ (x) and lim x ± φ (x) = ± q. We know from Section in [6], the function U(z) satisfies the equation (5) ivu + 1 U + (ω f( U )U =, while the functions Φ(z) and Θ(z) satisfy (6) Φ (Θ ) Φ + vθ Φ + (ω f(φ )Φ =, (Θ Φ vφ ) =. Integrating the second equation of (6), we have (7) Θ = vφ + g z dξ Φ, Θ(z) = vz + g Φ (ξ). where g is an integral constant. Thus, we obtain (8) Φ + (ω f(φ ))Φ + v Φ 4 g Φ 3 =, which is equivalent to the two dimensional Hamiltonian system (9) dφ dz = y, dy dz = g (ω + v )Φ 4 + f(φ )Φ 4 Φ 3

3 EXACT DARK SOLITON AND CHAOS WAVE 163 with the Hamiltonian (1) H(Φ, y) = 1 y + 1 Φ (ω + v )Φ + g Φ f(s) ds. For the cubic-quintic NLS with f(q) = αq + βq, (α, β) R, we have (11) H(Φ, y) = 1 y + 1 (ω + v )Φ + g Φ 1 αφ4 1 3 βφ6. It is easy to see that (9) is a singular traveling system with the singular straight line Φ = (see [4, 5]). Because even for the cubic-quintic NLS with f(q) = αq + βq, to our knowledge, the exact traveling wave solutions defined by (3) and the dynamical behavior of the unperturbed and perturbed traveling wave system (9) have not been studied. So, in this paper we shall consider these problems. Our studies show that the assumption there are kink modes of () (given in Section of [6]) is incorrect (see Remark 1 in Section below). Moreover, we obtain different results from [6], by using the Melnikov method to discuss the homoclinic bifurcations of the traveling wave systems for the perturbed equation (1) and equation (1) iu t = 1 u xx + f( u )u + ɛ(p (u) + V (x)u), where P (u) = u(δ +γ 1 u +γ u 4 ) is given by [3, page 111] and V (x) is a periodic function, and we have more interesting and rigorously proved results. The rest of this paper is divided into four sections. In Section, the dynamical behaviors of system (9) with f(q) = αq are studied, and exact parametric representations of the traveling wave solutions of () are given. In Section 3, the dynamical behaviors of system (9) with f(q) = αq + βq are studied, and exact parametric representations of the traveling wave solutions of () are devived. In Section 4, we consider the persistence of dark solition for the perturbed cubic-quintic NLS equation (1) without the term V (x)u. In Section 5, by using known Melnikov method, we investigate the chaotic dynamics of traveling wave solutions for the perturbed system (1). The exact solutions of () for the cubic NLS equation with f(q) = αq In this section, we consider equation (9) with f(q) = αq. It

4 164 JIBIN LI is known (see [4, 5]) that system (9) has the same invariant curves as its associated regular system (13) dφ dζ = yφ3, dy dζ = g (v + ω)φ 4 + αφ 6, where dz = Φ 3 dζ, for Φ. The first integral of (13) is (14) H(Φ, y) = 1 y + 1 (ω + v )Φ + g Φ 1 αφ4. Let Q(p) = αp 3 (v + ω)p + g. Clearly, Q (p) = 6αp (v + ω)p. It follows that the function Q(p) has at most two positive zeros for any parameter group (ω, v, g). We assume that α >, v + ω > and g < (v +ω) 3 7α. Then, function Q(p) has exact two positive zeros at p 1 and p with p 1 < p v +ω 3α < p. It means that system (13) has two positive equilibrium points E 1 ( p 1, ) and E ( p, ) in the right half phase plane. Let M(φ i, ) be the coefficient matrix of the linearized system of (13) at an equilibrium point (Φ j, ), j = 1,. Then we have J(Φ j, ) = det M(Φ j, ) = 4Φ4 j 3α (p p j ), j = 1,. Thus, by using the results of the qualitative theory of differential equations, we know that the equilibrium E 1 is a center, while the equilibrium E is a hyperbolic saddle point. The phase portrait of system (13) is shown in Figure 1. Let h 1 = H( p 1, ), h = H( p, ), where H(Φ, y) is given by (14). We see from Fig.1 that in the right half phase plane, for h (h 1, h ), a closed branch of the level curves defined by H(Φ, y) = h gives rise to a periodic solution of (13). For h = h, a branch of the level curves defined by H(Φ, y) = h gives rise to a homoclinic orbit of (13). Remark 1. It can be seen from Figure 1 that the straight line Φ = partitions all phase orbits of (9) into two symmetric parts in the left and right phase plane, respectively. An orbit in the right phase plane can not pass through the straight line Φ =. System (9) can not have a heteroclinic orbit connecting the point ( p, ) and ( p, ). Thus, it is not reasonable to assume that there is a kink solution φ (x) of (9) in section of [], such that φ ( x) = φ (x) and lim x ± φ (x) = ± q. Therefore, the corresponding Theorem.1 in [6] needs to be modified.

5 EXACT DARK SOLITON AND CHAOS WAVE 165 y x 1 FIGURE 1: The phase portrait of (13) when α >, v + ω > and g < (v +ω) 3 7α. Next, we consider the exact parametric representations for some bounded orbits of (9). 1. When h (h 1, h ), we know from (14) that y = g + hφ (v + ω)φ 4 + αφ 6 Φ = α(r 1 Φ )(r Φ )(Φ r 3 ) Φ. Thus, by using the first equation of (9), we have the parametric representation of the periodic orbit of (9) as follows (see [1]): (15) Φ(z) = (r 3 + (r r 3 )sn (Ωz, k)) 1, where k = r r 3 r 1 r 3, Ω = α(r 1 r 3 ). In addition, we see from the second formula of (7) that (16) z Θ(z) = vz + g dξ r 3 + (r r 3 )sn (Ωξ, k) = vz + g r 3 Ω Π(arcsin(sn(Ωz, k)), α 1, k), where α 1 = r3 r r 3, Π(,, ) is the elliptic integral of the third kind.

6 166 JIBIN LI. Letting k 1, we have from (15) the parametric representation of the homoclinic orbit of (9) as follows: (17) Φ(z) = (p m + (p p m )tanh (Ω z)) 1 = (p (p p m )sech (Ω z)) 1, where Ω = α(p p m ) and (p m, ) is the intersection point of the homoclinic orbit and the positive Φ-axis. Thus, we know from the second formula of (7) that (18) Θ(z) = vz + g = ( v + z dξ p (p p m )sech (Ω ξ) g ) z + g p p m p Ω p Ω p m ( p p m arctan tanh(ω z) p m To sum up, by using (3), we obtain the following result. Proposition 1. Suppose that the parameter conditions (I): α >, v + ω > and g < (v +ω) 3 7α hold. ). (1) Let Φ(z) and Θ(z) be defined respectively by (17) and (18). Then, equation () with f(q) = αq has the exact dark soliton solution (19) u(x, t) = Φ(z)e i(θ(z) ωt). () Let Φ(z) and Θ(z) be defined respectively by (15) and (16). Then, equation () with f(q) = αq has the exact periodic solution family given by (19). (3) In addition, let v and keep the conditions in (I). Then, when Φ(z) and Θ(z) are defined respectively by (17) and (18) with v =, (19) gives rise to a bubble solution of () with f(q) = αq. 3 The exact solutions of () for the cubic-quintic NLS equation with f(q) = αq + βq In this section, we consider equation (9) with f(q) = αq + βq. In this case, system (9) has the same invariant curves as its associated regular system () dφ dζ = yφ3, dy dζ = g (v + ω)φ 4 + αφ 6 + βφ 8,

7 EXACT DARK SOLITON AND CHAOS WAVE 167 where dz = Φ 3 dζ, for Φ. Let Q 1 (q) = βq 4 + αq 3 (v + ω)q + g. We have Q 1 (q) = q[4βq + 3αq (v + ω)]. Assume that β >, v + ω >. Then, we have = 9α + 16β(v + ω) >. Function Q 1(q) has a positive zero at. It follows that when g < (α + βq )q 3, function Q 1(q) has exact two positive zeros q 1 and q satisfying q 1 < q < q. It implies that system () has two positive equilibrium points E 1 ( q 1, ) and E ( q, ) in the right half phase plane. By using the results of the qualitative theory of differential equations, we know that the equilibrium E 1 is a center, while the equilibrium E is a hyperbolic saddle point. The phase portrait of system () is the same as that shown in Figure 1. q = 1 8β [ 3α+ ]. Hence, we have Q 1 (q ) = g βq 4 αq3 Let h 1 = H( q 1, ), h = H( q, ), where H(Φ, y) is given by (11). We see from Figure 1 that in the right half phase plane, for h (h 1, h ), a closed branch of the level curves defined by H(Φ, y) = h gives rise to a periodic solution of (). For h = h, a branch of the level curves defined by H(Φ, y) = h gives rise to a homoclinic orbit of (). 1. When h (h 1, h ), we have from (11) that (1) y = g + hφ (v + ω)φ 4 + αφ 6 + βφ 8 Φ = β(r 1 Φ )(r Φ )(Φ r 3 )(Φ r 4 ) Φ. Thus, by using the first equation of (9), we have the parametric representation of the periodic orbit of (9) as follows: () Φ(z) = ( ) 1 r 3 r 4 r 4 +, 1 α sn (Ω 1 z, k 1 ) where k 1 = α (r 1 r 4 ) r 1 r 3, α = r r 3 r r 4, Ω 1 = β(r 1 r 3 )(r r 4 ).

8 168 JIBIN LI (3) In addition, we know from the second formula of (7) that Θ(z) = vz + g = = ( v + ( v + where α 3 = r 4 α /r 3. z g r 4 Ω 1 g r 3 Ω 1 dξ r 3 r 4 1 α sn (Ω 1z,k 1) z r 4 + ) z g(r 3 r 4 ) r 3 r 4 Ω 1. When h = h, we have that dξ 1 α 3 sn (Ω 1 ξ, k 1 ) ) z g(r 3 r 4 ) r 3 r 4 Ω 1 Π(arcsin(sn(Ω 1 z, k)), α 3, k 1), y = g + h Φ (v + ω)φ 4 + αφ 6 + βφ 8 Φ = β(q Φ ) (Φ q m )(Φ q 4 ) Φ. Further, we obtain from the first equation of (9) that the parametric representation of the homoclinic orbit of (9) is given by (4) Φ(z) = ( q (q q m )(q q 4 ) (q m q 4 )cosh(ω o z) + (q q m q 4 ) ) 1, where Ω o = β(q q m )(q q 4 ) and (q m, ) is the intersection point of the homoclinic orbit and the positive Φ axis, q 4 < < q m < q. Hence, we know from the second formula of (7) that (5) Θ(z) = vz + g z q dξ (q q m)(q q 4) (q m q 4)cosh(Ω oz)+(q q m q 4) = vz + g q z [ (q (q m + q 4 ) q m q 4 q ] 1 dξ q (q m q 4 ) cosh(ω o ξ) + q (q m + q 4 ) q m q 4 = ( v + arctan g ) z + g (q q m )(q q 4 ) q 4 q m q Ω o q Ω o ( q 4 (q q m ) q m (q q 4 ) tanh(ω oz) ).

9 EXACT DARK SOLITON AND CHAOS WAVE 169 To sum up, by using (3), we obtain the following conclusion. Proposition. Suppose that the parameter conditions (II): β >, v + ω > and g < (α + βq )q 3 hold. (1) Let Φ(z) and Θ(z) be defined respectively by (4) and (5). Then, equation () with f(q) = αq + βq has the exact dark soliton solution (19). () Let Φ(z) and Θ(z) be defined respectively by () and (3). Then, equation () with f(q) = αq + βq has the exact periodic solution family given by (19). (3) In addition, let v and keep the conditions (II). Then, when Φ(z) and Θ(z) are defined respectively by () and (3) with v =, (19) gives rise to a bubble solution of () with f(q) = αq + βq. 4 The persistence of dark solition for the perturbed cubicquintic NLS equation (1) without the term V (x)u In this section, we consider the perturbed cubic NLS equation (6) iu t = 1 u xx + (α u + β u )u + ɛu(δ + γ 1 u + γ u 4 ). Considering the black solution given by (19) in section and writing Φ(z) v= = φ(x), we have the system (7) dφ dx = y, dy dx = g ωφ 4 + (αφ + βφ 4 )φ 4 φ 3 + ɛφ(δ + γ 1 φ + γ φ 4 ). Under the perturbation of (7), we shall consider the persistence of the homoclinic orbit φ (x) of the unperturbed system (7) ɛ= defined by (17) and (4), respectively. By the Melnikov s theory (see [6], section 4.8, Theorem 4), we compute the following Melnikov integral (8) M(δ, γ 1, γ ) = y (x)φ (x)(δ + γ 1 φ (x) + γ φ 4 (x))dx [ 1 = δφ (x) γ 1φ 4 (x) + 1 ] 6 γ φ 6 (x).

10 17 JIBIN LI where φ (x) is given by (17) or (4), y (x) = dφ (x)/dx. Substituting respectively (17) and (4) into (8), we obtain (9) M 1 (δ, γ 1, γ ) = δ(p p m ) + 1 γ 1(p p m ) γ (p 3 p3 m ), M (δ, γ 1, γ ) = δ(q q m ) + 1 γ 1(q q m) γ (q 3 q 3 m). For a given parameter pair (γ 1, γ ), taking respectively (3) δ = δ 1 3γ 1(p + p m ) + γ (p + p p m + p m ), 6 δ = δ 3γ 1(q + q m ) + γ (q + q q m + qm ), 6 we have M j (δ j, γ 1, γ ) =, j = 1,. Notice that Mj(δ,γ1,γ) δ, j = 1,. Thus, Theorem 4 of Section 4.8 in [7] gives the following result. Theorem 3. Under the assumptions (I) and (II) of Propositions 1 and, for sufficiently small ɛ and given parameter pair (δ 1, δ ), there exists δ = δ j + O(ɛ) such that system (9) with f(q) = αq and f(q) = αq+βq has a unique homoclinic orbit with the parametric representation φ ɛ (x) in an O(ɛ) neighborhood of the homoclinic orbit given by φ (x). Letting Θ ɛ (x) = x result. ds φ ɛ (s) in Theorem 3, then we have the following Corollary 4. For a sufficiently small ɛ, equation (6) has a unique dark soliton solution u(x) = φ ɛ (x)e i(θɛ(x) ωt). 5 Chaotic behavior of the traveling wave solutions for the perturbed cubic-quintic NLS equation (1) We now investigate the equation (31) iu t = 1 u xx + (α u + β u )u + ɛu(δ + γ 1 u + γ u 4 + V (x)), where the function V (x) is a periodic function, for example, V (x) = γ 3 sin(ϖx). Considering the black solution given by (19) in Section

11 EXACT DARK SOLITON AND CHAOS WAVE 171 and writing Φ(z) v= = φ(x), we have the system (3) dφ dx = y, dy dx = g ωφ 4 + (αφ + βφ 4 )φ 4 φ 3 + ɛφ(δ + γ 1 φ + γ φ 4 + V (x)). In this case, for every sufficiently small ɛ, the hyperbolic fixed points ( p, ) and ( q, ) of the unperturbed system (3) ɛ= become hyperbolic periodic orbits (see [, 8]). Under the perturbation of (3), we shall consider the existence of the transverse homoclinic points of the Pioncare map defined by the perturbed system (3). By using the Melnikov method (see [, 8]), along the homoclinic orbit φ (x) of the unperturbed system (3) ɛ= defined by (17) and (4), respectively, to compute the Melnikov functions, we have (33) M(δ, γ 1, γ, x ) = y (x)φ (x)(δ + γ 1 φ (x) + γ φ 4 (x) + V (x + x )) dx [ 1 = δφ (x) γ 1φ 4 (x) γ φ 6 (x) ] + φ (x)v (x + x ) φ (x)v (x + x ) dx. Letting V (x) = γ 3 sin(ϖx) and substituting respectively (17) and (4) into (33), we obtain M 3 (x ) = (δ(p p m ) + 1 γ 1(p p m) (34) γ (p 3 p3 m )) γ 3I 1 cos(ϖx ), M 4 (x ) = (δ(q q m ) + 1 γ 1(q q m ) γ (q 3 q 3 m)) γ 3 I cos(ϖx ),

12 17 JIBIN LI where I 1 = (p p m ) I = 4(q q m )(q q 4 ) cos(ϖx) dx cosh (Ω x) = (p p m )πϖ csch(ω πϖ), Ω = (q q 4 )Q π sin(ϖ sinh 1 Q ) 1 + Q sinh(πϖ/) cos(ϖx) dx (q m q 4 ) cosh(ω o x) + (q q m q 4 ) with Q = q q4 q m q 4. We see from (34) that if the conditions (35) γ 3 I 1 δ(p p m ) + 1 γ 1(p p m) γ (p 3 p3 m) > 1, γ 3 I δ(q q m ) + 1 γ 1(q q m) γ (q 3 q3 m) > 1 are satisfied, respectively, then the Melnikov functions M 3 (x ) and M 4 (x ) have simple zeros. It implies the existence of the transverse homoclinic points of the Pioncare map corresponding to the hyperbolic periodic points of the perturbed system (3). Thus, we have the following theorem. Theorem 5. Under the conditions (I) and (or) (II), taking V (x) = γ 3 sin(ϖx), if the conditions (35) hold, the traveling wave solutions defined by (3) of equation (31) possess chaotic dynamics in the sense of existence of Smale horseshoe in the Pioncare map of (3). Finally, we assume that δ = γ 1 = γ =, γ 3, then (31) is reduced to the Gross-Pitaevskii equation (1). We have from (34) that (36) M 3 (x ) = γ 3 I 1 cos(ϖx ), M 4 (x ) = γ 3 I cos(ϖx ). This means that M 3 (x ) and M 4 (x ) always have simple zeros. Hence, we have the following corollary. Corollary 6. Under the conditions (I) and (or) (II), taking V (x) = γ 3 sin(ϖx), then the traveling wave solutions defined by (3) of equation (1) possess chaotic dynamics in the sense of existence of Smale horseshoe in the Pioncare map of (3).

13 EXACT DARK SOLITON AND CHAOS WAVE 173 REFERENCES 1. P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Springer, Berlin, J. Guckenheimer and P. J. Holmes, Nonlinear Oscillationa, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, Berlin, Yu. S. Kivshar and D. Luther-Davies, Dark optical solitons: Physics and applications, Phys. Rep. 98 (1998), Jibin Li and Huihui Dai, On the Study of Singular Nonlinear Traveling Wave Equations: Dynamica System Approach, Science Press, Beijing, Jibin Li and Guanrong Chen, On a class of singular nonlinear traveling wave equations, Int. J. Bifur. Chaos 17 (7), D. E. Pelinovsky and P. G. Kevrekidis, Dark solitons in external potentials, Z. Angew. Math. Phys. 59 (8), L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New Nork, 199. Department of Mathematics, Zhejiang Normal University Jinhua, Zhejiang, 314 P. R. China School of Science, Kunming University of Science and Technology, Kunming, Yunnan, 6593 P. R. China address: jibinli@gmail.com or lijb@zjnu.cn

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