Periodic dynamics of a derivative nonlinear Schrödinger equation with variable coefficients

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1 ARTICLE MANUSCRIPT Periodic dynamics of a derivative nonlinear Schrödinger equation with variable coefficients Qihuai Liu a, Wenye Liu a, Pedro J. Torres b and Wentao Huang c a School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, China; b Departamento de Matematica Aplicada, Universidad de Granada, Granada, Spain; c School of Computing Science and Mathematics, Guilin University of Aerospace Technology, Guilin, China ARTICLE HISTORY Compiled July 4, 18 ABSTRACT We study the existence and multiplicity of periodic waves with nontrivial phase on the derivative nonlinear Schrödinger equation with periodic coefficient. The existence of infinitely many periodic solutions with nontrivial phase is proved by using Poincaré-Birkhoff twist theorem and the method of averaging. The sequence of rotation numbers for large amplitude periodic solutions tends to infinity, while the one for small amplitude periodic solutions tends to a certain constant. Additionally, exact expressions of small amplitude periodic solutions are obtained by introducing a small parameter. KEYWORDS Derivative nonlinear Schrödinger equation; variable coefficient; coherent state; periodic solution; nontrivial phase 1. Introduction The nonlinear Schrödinger equation NLS with cubic nonlinearity, also called Gross- Pitaevskii equation GPE, is an universally accepted model for many phenomena in Quantum Mechanics involving the propagation of wave packets, like the evolution of Bose-Einstein condensates BECs [1], propagation of electromagnetic waves in Nonlinear Optics [,3], dynamics of hot plasmas [4] and many others. Nevertheless, under some physical premises it becomes mandatory to consider enhanced versions of the GPE by introducing additional terms in order to deal with more realistic models. A known example is the derivative nonlinear Schrödinger equation DNLS. In the related literature, two types of DNLS are typically considered, namely the Chen-Lee- Liu equation CLLE [5] iψ t + ψ xx λ ψ ψ + iσ ψ ψ x = CONTACT qhuailiu@gmail.comq. Liu;ptorres@ugr.es P. J. Torres; huangwentao@163.com W. Huang.

2 and the Kaup-Newell equationkne [6] iψ t + ψ xx λ ψ ψ + iσ ψ ψ x =. In Nonlinear Optics, the derivative term models the self-steepening effect of short intense pulses [7]. The physical realization of CLLE has been reported through the interplay of quadratic and cubic nonlinearities in the frequency-doubling crystal [11]. On the other hand, KNE is used to describe the weakly dispersive Alfvén waves under the one-dimensional approximation in the magnetized plasmas [1]. Both equations are completely integrable by the inverse scattering transform and there exists an infinite number of conserved quantities. In fact, it is well-known that both equations are equivalent by gauge transformations [13]. There is a considerable body of work see for instance [8 1] that focuses on the identification of exact solutions. Typically such solutions fail to be preserved when we consider variable coefficients due to inhomogeneities on the medium. The main aim of this paper is to identify periodic coherent structures of the analogous DNLS equation with variable coefficient and iψ t + ψ xx λx ψ ψ + iσ ψ ψ x = 1 iψ t + ψ xx λx ψ ψ + iσ ψ ψ =. x In a Bose-Einstein condensate, the function λx models a variable scattering length, while in propagation of optical pulse it represents a variable Kerr coefficient. Due to its physical interest, DNLS equations with variable coefficients have been studied in many papers from different perspectives, see [14 ], only to cite some of of them. From an analytic point of view, quasi-periodic dynamics including the existence of Cantor families of smooth quasi-periodic solutions of small amplitude has been established in [1] by using infinite dimensional KAM theory. Also, we can refer [16,,3] for related developments. Our point of view is different. By assuming that the coefficient λx is a periodic function, we shall prove the existence of periodic waves with nontrivial phase of 1 and with respect to both t and x. Such solutions are known as modulated amplitude waves, and have been studied from different methods in the inhomogeneous GPE [4 7]. The method of proof starts with a standard separation of phase and amplitude dynamics. Then the amplitude equation is studied separately. The main results include two aspects. On the one hand, based on Poincaré-Birkhoff twist theorem, we prove the existence of infinitely many large amplitude periodic solutions, and the sequence of rotation numbers for these large amplitude periodic solutions tends to infinity, see Theorem 3.1. In this case, we does not need the smoothness assumption on λx. On the other hand, infinitely many small amplitude periodic solutions are identified by the method of averaging. This method provides the exact asymptotic profile. Comparing with large amplitude periodic solutions, the sequence of rotation numbers for such small amplitude periodic solutions tends to a certain constant, see Theorem 4.1. Both small and large amplitude periodic solutions have nontrivial phases, which is an essential novelty with respect to the existing literature on DNLS. The rest of the paper is organized as follows. In Section, we introduce a coherent structure ansatz to deduce the evolution equations of amplitude and phase for equation

3 1, while it turns out that the evolution equations for are completely analogous. Moreover, the rotation number of a solution is introduced. In Section 3, firstly we state one of our main theorems for large amplitude periodic solutions with nontrivial phases in Subsection 3.1, see Theorem 3.1. The proof is done in two steps. First, in Subsection 3. we shall prove the existence of infinitely many large periodic solutions for the amplitude evolution equation by Poincaré-Birkhoff twist theorem. Then, the proof of Theorem 3.1 is completed in Subsection 3.3 by a discussion of the associated rotation number. Section 4 is devoted to the existence of small amplitude periodic solutions by the method of averaging. Exact expressions up to a small parameter of these periodic solutions are presented, see Theorem 4.1. Finally, a concrete example is given to demonstrate how to find the asymptotically exact periodic solutions by applying Theorem 4.1, and a numerical simulation is shown.. Evolution equations of amplitude and phase.1. Coherent structure and evolution equations Consider a uniformly propagating coherent structure with the form ψx, t = Rx exp[iθx µt], 3 where Rx is the amplitude of the wave function, θx gives the phase dynamics, and µ is a constant. In the special background of BEC, µ denotes the chemical potential. When such a temporally periodic coherent structure 3 is also spatially periodic, it is called a modulated amplitude wave MAW, which has been widely studied [4,7,8] for the standard NLS. Substituting 3 into 1, we equate real and imaginary parts of the resulting equation, obtaining σr xr x + R xθ x + Rxθ x =, 4 µrx λxr 3 x σr 3 xθ x Rxθ x + R x =. 5 In view of 4 and noticing that d R xθ x + σ dx 4 R4 x = Rx σr xr x + R xθ x + Rxθ x =, the phase evolution equation becomes θ x = c R x σ 4 R x, 6 where c is an arbitrary constant of integration. Substituting 6 into equation 5, we obtain the amplitude evolution equation R x + µ cσ Rx c R 3 x λxr3 x + 3σ 16 R5 x =. 7 3

4 When the coefficients λx =, σ =, c and µ >, equation 7 is the Ermakov- Pinney equation whose solutions are expressed explicitly by Rx = c R x + µrx R 3 =, 8 x ρ cos µx + φ + 1 c µρ 4 sin 1 µx + φ, 9 which are all π/ µ-periodic, where ρ ρ, φ are arbitrary integral constants. On the other hand, when we take c =, equation 7 is a Duffing equation with cubic-quintic nonlinearity. It is interesting to remark that we can arrive to the same formulation starting from a NLS with cubic-quintic nonlinearity see for instance [9, 3] and the references therein, so our results are also applicable to this context. Along this paper, we always take the nonzero integral constant c. Performing the same operation of above, we obtain the evolution equations of amplitude and phase for as the following θ x = c R x 3σ 4 R x, 1 R x + µ + cσ Rx c R 3 x λxr3 x + 3σ 16 R5 x =. 11 As we can see that, comparing with 6 and 7, the difference is only that the coefficients differ by a constant, but this fact does not affect the main results. From now on, we consider 1 for concreteness, but it is important to remark that the conclusions are also valid for Eq.. We remark that, one may transform the solutions of 1 with periodic coherent structure 3 to the solutions of by the gauge transformation, however the gauge transformation may be not periodic with respect to t and x... Rotation number As usual, given a solution ψ of 1 with a positive and periodic amplitude, we define the associated rotation number by rotψ = θx x + x. 1 In view of 6, we know that the rotation number is well defined. Furthermore, we have that rotψ = x + 1 x x Now it is not difficult to obtain the following theorem. c R ξ σ 4 R ξ dξ. 13 4

5 Theorem.1. The rotation number rotψ is continuous with respect to the integral constant c on, or, +. Moreover, If σ < resp. σ >, then rotψ = + resp. rotψ =. c + c Proof. For c, we assume that Rx, c is a continuous branch of solutions of equation 7. Then the parametric integral in 13 is continuous in c, which implies that rotψ is continuous with respect to c. If σ <, then for c, +, by using geometric inequality we have Together with 13, it follows that rotψ = c + c R ξ, c σ 4 R ξ, c c σ. 1 x x c σ = +. c + x + c + The case σ > can be studied analogously. c R ξ, c σ 4 R ξ, c dξ In a special case, the calculation of the rotation number can be simplified. If Rx is a T -periodic solution of 7, rotψ = n + 1 nt n 1 k+1t k= kt c R ξ σ 4 R ξ dξ, which yields that rotψ = 1 T T c R ξ σ 4 R ξ dξ Infinitely many large amplitude periodic solutions Along this section, we always assume λx is a continuous T -periodic function. We shall prove the existence of infinitely many large amplitude periodic solutions of 1 both with respect to t and x by using qualitative theory of ODE and fixed point theorems. Periodic solutions of large amplitude in this context means that the supremum of the modules of solutions sup ψt, x is very large Main result for large amplitude periodic solutions Theorem 3.1. Assume that λx is a continuous periodic function with the least positive period T and σ. Then for any constant µ and any positive integer m, there exists K m > such that, for any positive integer k > K m, equation 1 has a sequence of periodic solutions {ψ n,k x, t} n with the form ψ n,k x, t = 5

6 R n,k x exp[iθ n,k x µt], which is periodic with respect to x with the least period mt such that the notation number is given by and for σ <, rotψ n,k = π mt k, while for σ >, rotψ n,k = π mt k, sup n + x [,mt ] ψ n,k x, t = +, n + inf ψ n,kx, t =. x [,mt ] The condition that the least positive period of λx is T, is only used to guarantee that the least positive period of ψ n,1 x, t on x is also T. The proof of Theorem 3.1 includes two steps. Firstly, for any positive integer m, we shall prove in Subsection 3. the existence of infinitely many mt -periodic solutions for the amplitude evolution equation 7. Then in Subsection 3.3, we shall complete the proof by discussing the rotation numbers. 3.. Periodic dynamics of the amplitude evolution equation For convenience of estimating the rotation angle, we translate equation 7 with a constant so that it becomes R x+ µ cσ Rx+c c Rx + c 3 λxrx+c 3 + 3σ 16 Rx+c 5 =, 15 where c = 16c /3σ 1/8 is a rest point of the Ermakov- Pinney equation 8. Now, let us consider the equivalent system R x = Sx S x = 3σ c 16 R5 x + Rx + c 3 + gλx, Rx, 16 where gλ, R is the remainder given by gλ, R = λr + c 3 µ cσ R + c 3σ R + c 5 R Equation 15 is a specific Duffing equation with superlinear potential and a strong singularity. The energy near a strong singularity becomes infinity and this fact is helpful for obtaining either the a priori bounds needed for the application of degree theory, or the fast rotation needed in applying the Poincaré-Birkhoff theorem. When one side of the potential has a strong singularity and the other side is superlinear, the existence of infinitely many positive periodic solutions for second order Duffing equations has been proved by Del Pino and Manásevich via Poincaré-Birkhoff theorem [31]. The result was also extended to the case of semilinear behaviour and strong singularity by Wang and Ma, see [3]. Moreover, multiplicity of positive periodic solutions for superlinear singular equations has been obtained in [33] based on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem. In our case, because of 6

7 the specificity of 15, the proof of the existence of infinitely many periodic solutions is similar to [31] Basic lemmas Let Rx, Sx = Rx, R, S, Sx, R, S be the solution of 16 satisfying the initial value where R + c. R, R, S = R, S, R, S = S, 17 Lemma 3.. There exists a unique solution Rx, Sx of 16 with initial conditions 17, and it is defined on the whole x-axis. Proof. Since the system is locally Lipschitz continuous with respect to the unknown function R, uniqueness of solutions is obtained. By contradiction, we assume Rx, R, S, Sx, R, S is a non-continuable solution of 16 which is defined on α, β with < α < β < +. Define the function W : α, β R by Then, we have W x = 1 S x + 1 c Rx + c + σ 3 R6 x. dw dt x = gλx, RxSx C Sx R 4 x + C 1 C W x + C 1, 16 where the last inequality follows from the inequality y α1 1 yα yαn n α 1 y 1 + α y + α n y n, for all y i, α i >, i = 1,,, n such that n i=1 α i = 1. By using Grownwall inequality, we obtain that, for x [x, β, and for x α, x, M e cβ x M 1 W x M e cβ x M 1 M e cα x M 1 W x M e cα x M 1. where the constants M = W x + C 1 /C and M 1 = C 1 /C. Therefore, we know that Sx < + and, c < Rx < + or < Rx < c, x α, β, which implies that the solution can be continuable both at α and β. Thus we arrive at a contradiction and end the proof. According to Lemma 3., the Poincaré map P : c, + R R is well defined by P : R, S R 1, S 1 = RT, R, S, ST, R, S. 7

8 Obviously, fixed points of the Poincaré map P correspond to T -periodic solutions of equation 16. In order to describe the position of a given orbit Rx, Sx of system 16, we introduce a function H : c, + R R by HR, S = 1 S + 1 c R + c + σ 3 R6. Lemma 3.3. For any positive constants T and h 1, there exists h > sufficiently large such that, for HR, S h, HRx, Sx h 1, for all x [, T ], where Rx, Sx is the solution of 16 with the initial value point R, S. Moreover, if HR, S = h, then for all x [, T ], h + M 1 e ct M 1 HRx, Sx h + M 1 e ct M 1, where M 1 is a positive constant. Proof. From the proof of Lemma 3., we know that, for all x [, T ], HR, S + C 1 e ct C 1 HRx, Sx HR, S + C 1 e ct C 1, C C C C where C i, i =, 1 are certain constants. Take M 1 = C 1 /C and h = h 1 + M 1 e ct M 1. Then for HR, S h, we have HRx, Sx h 1, x [, T ]. To depict the orbit Rx, Sx of 16, we take the polar coordinates φ, ρ R, S = ρ cos φ, ρ sin φ defined on a half plane that excludes the origin. From Lemma 3.3, we know the solution of 16 with the initial value R, S such that HR, S h does not vanish in a finite time, that is, the polar form representation is well defined. Under this transformation, system 16 becomes dρ dx = 1 RS 3σ c ρ 16 R5 S + R + c dφ dx = 1 S ρ 3σ 16 R6 + 3S + Sgλx, R c R + c 3R + Rgλx, R Denote by ρx, φx = ρx, ρ, φ, φx, ρ, φ the solution of 18 satisfying the initial value ρ, φ = ρ, φ with R = ρ cos φ, S = ρ sin φ. Then we can rewrite the Poincaré map P as follows: P : ρ, φ ρ 1, φ 1 = ρt, ρ, φ, φt, ρ, φ,. 18 8

9 with R = ρ cos φ > c, S = ρ sin φ. Lemma 3.4. Assume T, K are given arbitrarily positive constants. For sufficiently large h 1 >, there exists h > sufficiently large such that, if HR, S h > h 1, then for all x [, T ], HRx, Sx h 1 and dφ dx 1 sin φ K cos φ <, x [, T ], 19 where ρx, φx is the solution of 18 with the initial value point ρ, φ. Proof. For any T > and h 1 >, by Lemma 3.3 there exists h > sufficiently large such that, for HR, S h, HRx, Sx = 1 S x + 1 c Rx + c + σ 3 R6 x h 1, x [, T ]. On the one hand, for any positive constant c, c and any constant R such that R + c >, the following inequality 3c R + c = 4c c R + c c 1 4 3R + c c c c 3R + c R + c c 4 5 c 3 = c c R + c c 4 6 c 3 R + c holds. On the other hand, by the inequality above and together with, we have which yields that Rgλ, R S 3σ c 16 R6 + R + c 3R =Rgλ, R S 3σ c 16 R6 + R + c c c R + c 3 Rgλ, R S 3σ 16 R6 R + c + 37 c 4 6 c 3 R + c 1 c = S + R + c + σ 3 R6 1 S gλ, R, c Rgλ, R S 3σ c 16 R6 + R + c 3R h 1 1 S gλ, R, 1 where g is a polynomial of order 6 given by gλ, R = 5σ 3 R6 Rgλ, R 37 c 4 6 c 3 R + c. 9

10 It is easy to see that g satisfies the superlinear condition gλ, R/ 1 R + as R + uniformly with respect to λ in any bounded set, which is equivalent to that, for any given K >, there exists a constant CK, λ > such that gλ, R K 1 R CK, λ, λ, R [ M, M] c, + with any positive constant M. Since λx is periodic, it is uniformly bounded but the bound changes from one λx to another. By using 18, 1 and, for any given K >, we take h > sufficiently large such that h 1 > CK, we know that Thus we complete the proof. dφ dx = 1 Rgx, ρ R S 3σ c 16 R6 + R + c 3R h 1 CK, λ ρ 1 sin φ K cos φ < A suitable estimate of the rotation time By the proof of Lemma 3.4, we know that the orbit ρx, φx of 18 turns clockwise around the origin. Using a dynamical perspective, if the independent variable x is regarded as the time, then we shall estimate the time at which the solution turns one round around the origin. Roughly speaking, solutions with high energy rotate very fast. Lemma 3.5. If ρx, φx is the solution of 18 with the initial value point ρ, φ such that and φx φx 1 = π, then Hρ cos φ, ρ sin φ = h x = x x 1 =. 3 h + h + Proof. By Lemma 3.3, for h large enough, we have HRx, Sx h 1 := h + C 1 e ct C 1, x [, T ]. 4 C C By Lemma 3.4, for any given T > and h large enough, we know that dφ dx 1 sin φ K cos φ <, x [x 1, x ] [, T ]. 1

11 Therefore, we have that π dφ π x = x x 1 sin φ + K cos φ = 4 dφ sin φ + K cos φ π = 4 π = 4π K. [ dξ 4 ]π cos ξ + K sin ξ = arctan K tan ξ K π Let h +, then we know that inf x Rx c, sup x Rx + and h 1 CK + with CK corresponding to any given K in. In this case, we can take K arbitrarily large. Then we have h + x x 1 Therefore, we complete the proof of this lemma. inf K,+ 4π K = Existence of infinitely many periodic solutions Theorem 3.6. Assume that λx is a continuous T -periodic function and σ. Then for any positive integer m, equation 15 has infinitely many periodic solutions R j x with the least positive period mt such that j + Proof. Consider the function sup R jx + x mt 1 R j x + c + R6 j x = +. 5 m ρ, φ = φmt, ρ, φ φ, ρ, φ, where ρ, φ Ω := {ρ, φ : Hρ cos φ, ρ sin φ h }, then m ρ, φ is continuous on Ω. Taking a properly large constant a 1, there exists a positive prime number q 1 such that } inf { m ρ, φ : Hρ cos φ, ρ sin φ = a 1 > q 1 π. Therefore, when ρ, φ is such that Hρ cos φ, ρ sin φ = a 1, we have φmt, ρ, φ φ, ρ, φ > q 1 π. 6 By Lemma 3.5, there exists a constant b 1 > a 1 such that φmt, ρ, φ φ, ρ, φ < q 1 π. 7 if Hρ cos φ, ρ sin φ = b 1. 11

12 Construct an annular domain by A 1 = The m-order iteration of the Poincaré map {ρ, φ : a 1 Hρ cos φ, ρ sin φ b 1 }. P m : ρ, φ ρ mt, ρ, φ, φ mt, ρ, φ of 18 is an area-preserving map. In view of 6 and 7, we know that P m is a twist map on the annular domain A 1. By Poincaré-Birkhoff twist theorem, P m has at least two fixed points ρ i, φ i A 1 i = 1, such that φmt, ρ i, φ i φ, ρ i, φ i = q 1 π i = 1,, 8 which are equivalent to two mt -periodic solutions ρx, ρ i, φ i, φx, ρ i, φ i i = 1, of 18. If m = 1, since λx is a periodic function with the least positive period T, the least positive period of ρx, ρ i, φ i, φx, ρ i, φ i i = 1, also is T. When m > 1, we follow the proof in [34]. By contradiction, we assume the least positive period of ρx, ρ i, φ i, φx, ρ i, φ i i = 1, is nt with < n < m. Then ρ i, φ i i = 1, are the n-period point of P m and n is the least positive period. Let m = sn + q, s, q Z such that s 1 and q < n. Since P m ρ i, φ i = ρ i, φ i and P n ρ i, φ i = ρ i, φ i, we have P q ρ i, φ i = ρ i, φ i which yields q =. Therefore, m = ns and s > 1. On the other hand, the number of turns of periodic solutions ρx, ρ i, φ i, φx, ρ i, φ i at one period nt must be a integer N. By Lemma 3.5, N > 1 if a 1 is large enough. Then the number of turns of periodic solutions ρx, ρ i, φ i, φx, ρ i, φ i at the time interval [, mt ] is sn. From 8, we know that q 1 = sn with s > 1 and N > 1. Since q 1 is a prime number, this is a contradiction. Analogously, we can construct infinitely many disjoint annular domains A j = {ρ, φ : a j Hρ cos φ, ρ sin φ b j }, j = 1,,, with a j+1 > b j j = 1,, and b j + as j +. Then we obtain infinitely many periodic solutions ρ j x, φ j x with the least positive period mt such that sup Hρ 1 cos φ 1, ρ 1 sin φ 1 < < sup Hρ j cos φ j, ρ j sin φ j < +. x mt x mt Therefore, such periodic solutions are distinct and j + sup x mt 1 R jx + 1 which implies 5. The proof is done. 1 R j x + c + σ 3 R6 j x = +, 1

13 3.3. Proof of Theorem 3.1 By Theorem 3.6, for any c and any positive integer m, equation 15 has infinitely many periodic solutions R n x; c with the least positive period mt such that sup n + x [,mt ] R n x; c = +, n + inf R nx; c = c. x [,mt ] Solutions of the amplitude evolution equation 7 are recovered from 15 by adding a constant c. Therefore, for any c and any positive integer m, equation 7 has infinitely many periodic solutions R n x; c with the least positive period mt such that sup n + x [,mt ] R n x; c = +, n + inf R nx; c =. x [,mt ] To prove that ψ n,c x, t = R n x; c exp[iθ n x; c µt] is mt -periodic with respect to x, we need to prove that exp[iθ n x; c] is also a mt -periodic function. First, we consider the case σ <. In this case, we take the integrate constant c >. By Theorem.1 and 14, we have that rotψ n,c = c + c + 1 mt mt c R nξ; c σ 4 R nξ; c dξ = +. Moreover, rotψ n,c is continuous with respect to c on, +. Therefore, there exists K m > large enough such that, for any integer k > K m, there is a positive constant c k such that From 6, we know that θ n x; c k = x rotψ n,ck = π k. 9 mt c k R x; c k σ 4 R x; c k dx. Let θ n x; c k = x c k R x; c k σ 4 R x; c k rotψ n,ck dx, then we know that θ n x; c k is mt -periodic with respect to x. Therefore, we have exp[iθ n x; c k ] = exp[i θ n x; c k + rotψ n,ck x] = exp[i θ n x; c k ] cos[rotψ n,ck x] + i sin[rotψ n,ck x]. By 9, exp[iθ n x; c k ] is mt -periodic with respect to x. Therefore, we conclude that, when σ <, for any positive m, there exists K m > such that for all positive integer k > K m, equation 1 has infinitely many periodic solutions ψ n,k x, t with the form ψ n,k x, t := ψ n,ck x, t = R n x; c k exp[iθ n x; c k µt], 13

14 which is periodic with respect to x with the least period mt such that and sup n + x [,mt ] rotψ n,k = + k + ψ n,k x, t = +, n + inf ψ n,kx, t =. x [,mt ] When σ >, we take the integrate constant c <. In this case, rotψ n,c as c, and there exists large enough K m > such that, for any integer k > K m, there is a negative constant c k such that rotψ n,ck = π mt k. Then the rest proof is similar to the case σ < and we do not repeat it again. 4. Small amplitude periodic solutions In this section, we focus on the small amplitude periodic solutions of 1 by the method of averaging. In the following, we state our main result for small amplitude periodic solutions Main result Let us use some notations as follows T = π 1, λ = µ T f 1 φ = 1 T g 1 φ = 1 T T T T λxdx, λx sin φ + µxdx, λx cos φ + µxdx, f φ = 1 T g φ = 1 T T T λx sin 4φ + µxdx, λx cos 4φ + µxdx. Theorem 4.1. Assume that λx is a C 1 π/ µ-periodic function and σ. Assume that there exists a solution ρ, φ, 1/ 4 µ 1/ 4 µ, + R of the algebraic equations F 1 ρ, φ := µρ f 1 φ + µρ 4 1f φ =, F ρ, φ :=µρ 4 1 3µρ λ + µρ σ +4µ ρ 8 + 1g 1 φ + µ ρ 8 1g φ = such that F 1, F ρ, φ. 3 ρ,φ 14

15 Then, there exists a sequence of positive integer numbers {k n } n N and a positive real sequence {ɛ n } n N such that k n + and ɛ n as n + and for every n N, equation 1 has a k n π/ µ-periodic solution ψ µ,n x, t in x with the form ψ µ,n x, t = R µ,n x exp[iθ µ,n x µt] such that rotψ µ,n µ, as n +, where for all x tan µx + φ tanφ θ µ,n x = arctan µρ arctan µρ + Oɛ n, R µ,n x = ɛ n ρ [ π + φ µ, π φ ]. µ cos µx + φ + 1 µρ 4 sin µx + φ + Oɛn /3, This result is motivated by [6] see Theorem 1, where the author proves the existence of spatial periodic solutions with the coherent structure 3 for the Gross- Pitaevskii equation GPE in the context of Bose-Einstein condensates. Comparing with the result of [6], µ of 3 in [6] changes, say ψx, t = Rx exp[iθx µ n t]. For each sufficiently large µ n, there exists a spatial periodic solutions of GPE. On the contrary, in Theorem 4.1, we are fixing the value of µ. On the other hand, the detailed information and exact expression of periodic solutions has been established by Theorem 4.1. The sequence of solutions ψ µ,n t, x satisfies the uniformly its where θ x = arctan R x =ρ ψ µ,nt, x/ ɛ n = ψ t, x = R x exp[iθ x µ n t], n + tan µx + φ tanφ µρ arctan µρ, cos µx + φ + 1 µρ 4 sin µx + φ, x [ π + φ µ, π φ ]. µ 4.. Averaging of the amplitude evolution equation For convenience, we recall the amplitude evolution equation R x + µ cσ Rx c R 3 x λxr3 x + 3σ 16 R5 x =, 31 where c is an arbitrary constant. Let us introduce a small parameter ɛ > by the scale transformation Then, equation 31 becomes that Rx ɛrx, c = ɛ. R x + µrx 1 R 3 x ɛσ Rx ɛλxr3 x + ɛ 3σ 16 R5 x =. 3 15

16 Rewrite 3 in the equivalent form of one order R x = Sx S x = µrx + 1 R 3 x + ɛσ Rx + ɛλxr3 x ɛ 3σ 16 R5 x. 33 To apply the averaging theorem, the first step is to transform 3 into the standard form of averaging. Let us define a transformation Φ : R, S ρ, φ by R = ρ cos µx + φ + 1 µρ 4 sin µx + φ S = ρ 1 µ µρ 4 1 cos µx + φ sin µx + φ, cos µx + φ + 1 µρ 4 sin µx + φ which has been introduced in [5]. Under the transformation 34, system 33 becomes S ɛ σ V ρ R + ɛλxr3 ɛ 3σ 16 R5, ɛ σ R + ɛλxr3 ɛ 3σ R 16 R5 ρ, dρ dx = dφ dx = 1 V ρ where the function V :, 1/ 4 µ 1/ 4 µ, + R + is defined by V ρ = µρ + 1 ρ Then it follows that dρ dx = ɛ 1 8µ 3/ µρ 4 + 1λx + µρ σ sin φ + µx ρ +µρ 4 1λx sin 4φ + µx + Oɛ, dφ dx = ɛ 1 8µ ρ µρ 4 µρ 4 1 3µρ 4 + 1λx + µρ σ 1 + µ ρ 8 + 1λx + µρ σµρ cos φ + µx +µ ρ 8 1λx cos 4φ + µx + Oɛ. The averaging equation of order one corresponding to 36 is given by d ρ dx = ɛ 1 8µ 3/ µ ρ f 1 ρ φ + µ ρ 4 1f φ, d φ dx = ɛ 1 µ ρ 4 8µ ρ µ ρ 4 1 3µ ρ λ + µ ρ σ 1 +4µ ρ 8 + 1g 1 φ + µ ρ 8 1g φ,

17 where λ, f i φ and g i φ, i = 1,, have been defined in Subsection Proof of Theorem 4.1 Under the stated hypotheses, ρ, φ is a non-degenerate rest point of 37. By the averaging theorem, there exists a sufficiently small ɛ > such that, for all ɛ, ɛ, there is a π/ µ-periodic solution ρx, φx of 36 such that ρx, φx = ρ + Oɛ, φ + Oɛ. Using the transformation 34, we obtain a π/ µ-periodic solution Rx = ρ cos µx + φ + 1 µρ 4 sin µx + φ + Oɛ of 3. According to the scale transformation, we know there is a π/ µ-periodic solution Rx; ɛ, µ = ɛρ cos µx + φ + 1 µρ 4 sin µx + φ + Oɛ 3/ 38 of 31 where the integrate constant is taken by ɛ. In the following, we consider the solutions of the phase evolution equation with the integration constant ɛ θ x; ɛ, µ = ɛ R x; ɛ, µ σ 4 R x; ɛ, µ. Then it follows from 38 that, for x [ π/ µ φ / µ, π/ µ φ / µ ], θx; ɛ, µ = = x x = arctan ɛ R x; ɛ, µ σ 4 R x; ɛ, µ dx = ρ cos µx + φ + 1 µρ x ɛ R dx + Oɛ x; ɛ, µ sin µx + φ 1 dx + Oɛ tan µx + φ tanφ µρ arctan µρ + Oɛ, where we agree that arctantan±π/ = ±π/. Although we have an exact expression of θx up to order one of ɛ, we also need to prove that θx; ɛ, c is π/ µ-periodic. The idea for the proof of this claim is motivated by [6]. By 14, the rotation number is rotψ ɛ,µ = 1 T T ɛ R ξ; ɛ, µ σ 4 R ξ; ɛ, µ dξ + Oɛ π φ µ µ ɛ = π π + φ R ξ; ɛ, µ σ 4 R ξ; ɛ, µ dξ + Oɛ µ = µ + Oɛ. 17

18 Let us define I ɛ = {rotψ ɛ,µ : ɛ, ɛ }. Observe that rotψ ɛ,µ µ, as ɛ +. Therefore, µ belongs to the closure of I ɛ. At this moment, we distinguish two possibilities: Case I: I ɛ = { µ} for all ɛ, ɛ. In this case it is easy to arrive to the conclusion. Since θx; ɛ, µ = θx; ɛ, µ rotψ ɛ,µ x is π/ µ-periodic, ψx, t = Rx; ɛ, µ exp[i θx; ɛ, µ + rotψ ɛ,µ x µt] is π/ µ-periodic. Therefore, ψ µ,ɛ x, t is nπ/ µ-periodic. The result is proved by taking k n = n and ε n = 1/n for n sufficiently large, say n > n. Case II: the interval I ɛ is open. Since µ is in the closure of I ɛ and rotψ c,ɛ is continuous with respect to ɛ on, ɛ, at least one of the sequences µ + µ/n or µ µ/n, which is corresponding to a positive sequence {ɛ n } n N, belongs to I ɛ for n sufficiently large, say n > n. Suppose the first option holds, the second one being completely analogous. Then, let ψ µ,n x, t := ψ µ,ɛn x, t = R µ,n x exp[iθ µ,n x µt] with R µ,n x = Rx; ɛ n, µ and θ c,n x = θx; ɛ n, c. As in the proof of Theorem 3.1, we know that θ µ,n x = θ µ,n x rotψ µ,ɛn x is π/ µ-periodic with respect to x. Therefore, the solution of 1 ψ µ,n x, t = R µ,n x exp[i θ µ,n x + rotψ n,ɛn x µt] is k n π/ µ-periodic in x with k n = nn + 1. Moreover, rotψ µ,n µ as n +. Therefore, we complete the proof of Theorem 4.1. Remark 1. In the proof of Theorem 4.1, the important step is to prove the existence of periodic solutions of amplitude evolution equation, which is a continuation problem of periodic solutions for 3 with a small positive parameter ɛ. When ɛ =, the orbits of the unperturbed system fill up the whole phase space. Every orbit, which runs clockwise around the rest point 1/ 4 µ,, is closed and π/ µ-periodic. Under the perturbation with ɛ, these closed orbits may be destroyed or the period of the closed orbits may change. However, the averaging theorem guarantees that the nondegenerate orbit R x, S x = Rx, ρ, φ, Sx, ρ, φ defined by 34 persists, where ρ, φ is a non-degenerate zero of F 1, F, see Figure Numerical simulation Let us expand λx into a Fourier series λx = a + + a k cos k µx + b k sin k µx. k=1 18

19 4 R, S S x R x Figure 1. The orbits of the unperturbed system of 3. We assume that the Fourier series converges uniformly to λx on R. Taking T = π/ µ, λ = a /, then we have that f 1 φ = 1 T = T + ak T k=1 T + b k T λx sin φ + µxdx cos k µx sin φ + µxdx T sin k µx sin φ + µxdx = 1 a 1 sin φ + b 1 cos φ. Similarly, with the notations of f, g 1 and g defined in Subsection 4.1 we obtain that Then it follows that f φ = 1 a sin 4φ + b cos 4φ, g 1 φ = 1 a 1 cos φ b 1 sin φ, g φ = 1 a cos 4φ b sin 4φ. F 1 ρ, φ = µρ a 1 sin φ + b 1 cos φ + 1 µρ 4 1 a sin 4φ + b cos 4φ, F ρ, φ =µρ 4 1 3µρ a + µρ σ + µ ρ 8 + 1a 1 cos φ b 1 sin φ + 1 µ ρ 8 1 a cos 4φ b sin 4φ. To illustrate numerically the main result obtained in Subsection 4.1, we take σ = 1, a =, a 1 = 1, b 1 =, a = 1, b =, 19

20 so that F 1 and F become F 1 ρ, φ = sin φ µρ µρ 4 1 cos φ, F ρ, φ =µρ 4 1 3µρ In this case, λx is given by + µρ + µ ρ cos φ + 1 µ ρ 8 1 cos 4φ. λx = 1 + cos µx + cos4 µx. When φ = kπ k Z, we need to find the positive roots ρ 1 of equation hρ = µρ µρ ρ µ ρ =. Since hh+ <, by the zero-point theorem we know that, for all positive number µ 3 5, the function hρ has at least one positive root ρ = ρ, 1 1, +. The Jacobi determinant at the point ρ, is given by ρ, = F 1, F ρ, φ = 16µ ρ 5 µρ 4 11ρ ρ, We claim that ρ,. In fact, by contradiction, we assume ρ, =, then µ = ρ 4 11ρ Substituting µ into the function hρ, we obtain hρ = 363ρ ρ 4 + 6ρ + 8 ρ 11ρ + 3 <. which is a contradiction since ρ is a root of h. For example, when µ = 1, we find a numerical solution ρ, φ =.8866, of F 1 ρ, φ = and F ρ, φ =. The Jacobi determinant at the point ρ, φ is given by F 1, F ρ, φ = ρ,φ Then according to Theorem 4.1, there exists a sequence of positive integer numbers {k n } n N and a positive real sequence {ɛ n } n N such that k n + and ɛ n as n + and for every n N, equation 1 has a small amplitude k n π-periodic solutions ψ n x, t in x with the form ψ n x, t = R n x exp[iθ n x t], where m Z, tan x mπ θ n x = arctan ρ R n x = ɛ n ρ + mπ + Oɛ n, cos x + 1 ρ 4 sin x + Oɛn /3, x [ m 1π, m + 1π ].

21 RHxL Π Π 3Π.79 4Π x 15 ΘHxL 1 5 Π Π 3Π 4Π x a The contour picture of Reψ b The curves of R and θ Figure. The plots of the it function ψ t, x corresponding to the sequence of periodic solutions ψn t, x, where the parameters are taken by σ = 1, µ = 1 and λx = 1 + cos x + cos 4x. The sequence of solutions ψn t, x satisfies the uniformly its Rn x/ n R x and θn x θ x as n +. We have made a contour plot of the function ψ t, x = R x exp[iθ x t] which is π-periodic in x, see Figure a. The curves of θ and R have been depicted in Figure b. The difference of θn, Rn and θ, R is up to one order of n. In the case that φ = π + kπ k Z, we shall prove that there exists a positive root ρ 6= 1 of the algebraic equation 7 7 hρ = µρ 1 µρ ρ + + µ ρ8 + 1 =. 4 Since hh+ <, from the zero-point theorem we know that, for all positive number µ 6= 11 7, the function hρ has at least one positive root ρ = ρ, 1 π 1, +. The corresponding Jacobi determinant at the point ρ, is given by π F1, F ρ, = ρ, φ π ρ, 4 = 16µρ 3µρ ρ π π We claim that ρ, 6=. In fact, by contradiction, we assume ρ, =, 1 then µ = 3 1 ρ 4 ρ + 1. Substituting µ into the function hρ, we obtain hρ = 99ρ ρ4 + 18ρ ρ ρ <.

22 which is a contradiction since ρ is a root of h. For example, when µ = 1, we find a numerical solution ρ, φ = 1.179, π of F 1 ρ, φ = and F ρ, φ =. The Jacobi determinant at the point ρ, φ is given by F 1, F ρ, φ = ρ,φ Then according to Theorem 4.1, there exists a sequence of positive integer numbers {k n } n N and a positive real sequence {ɛ n } n N such that k n + and ɛ n as n + and for every n N, equation 1 has a small amplitude k n π-periodic solutions ψ n x, t in x with the form ψ n x, t = R n x exp[iθ n x t], where m Z, θ n x = arctan R n x = ɛ n ρ tan x mπ π ρ + m + 1 π + Oɛ n, sin x + 1 ρ 4 cos x + Oɛn /3, x [mπ, m + 1 π]. The sequence of solutions ψ n t, x satisfies the uniformly its R n x/ ɛ n R x and θ n x θ x as n +. Acknowledgments The authors wish to express their thanks to Professor Peter W. Bates, Michigan State University, for reading and improving this manuscript. This work is partially supported by NSFC Grant Nos , 11661, Guangxi Natural Science Foundation Nos. 17GXNSFFA1981, 16GXNSFDA3831, Innovation Project of Guet Graduate Education No. 18YJCX59 and Spanish MICINN Grant with FEDER funds MTM C-1-P. References [1] Carretero-González R, Frantzeskakis D, Kevrekidis PG. Nonlinear waves in Bose- Einstein condensates: physical relevance and mathematical techniques. Nonlinearity. 8; 17:R139 R. [] Kivshar YS, Agrawal G. Optical solitons: from fibers to photonic crystals. Academic Press; 3. [3] Agrawal GP. Nonlinear fiber optics. Academic Press; 7. [4] Malomed B. Nonlinear Schröinger Equations. in Scott A, Encyclopedia of Nonlinear Science, New York: Routledge. 5;: [5] Chen H, Lee Y, Liu C. Integrability of nonlinear Hamiltonian systems by inverse scattering method. Physica Scripta. 1979;: [6] Kaup DJ, Newell AC. An exact solution for a derivative nonlinear Schröinger equation. J Math Phys. 1978;194:

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