Orbital stability of periodic wave solution for Eckhaus-Kundu equation

In this paper, we mainly study the orbital stability of periodic traveling wave solution for the Eckhaus-Kundu equation with quintic nonlinearity, which is not a standard Hamilton system. Considering the studied equation is not a standard Hamilton system, the method presented by M. Grillakis and others for proving orbital stability cannot be applied directly, and this equation has two higher order nonlinear terms. So, by constructing three conserved quantities, using detailed spectral analysis and appropriate techniques, we overcome the complexity of the studied equation developed in calculation and proof, then, a conclusion on the orbital stability of the dn periodic wave solution for the Eckhaus-Kundu equation is obtained. As an extension of the proof for the above results, we also prove the orbital stability of the solitary wave for the studied Eckhaus-Kundu equation.


Introduction
It is well known that nonlinear phenomena exist in various fields of science and engineering, such as solid-state physics, biophysics, optical fibers, fluid dynamics, etc [1][2][3].With the development of nonlinear science, many nonlinear complex systems in these fields can be modeled by nonlinear evolution equations (NLEEs), and the well-known nonlinear Schrödinger-type equation (NLS) is one of the most critical models in NLEEs, which is often applied to space plasma, coastal engineering, nonlinear optics and so on.Accordingly, the study of nonlinear science has attracted the attention of a large number of researchers.Over the years, many effective methods were developed to study nonlinear problems.For example, there were some new advances on the solutions of nonlinear wave in recent years.Xu et al [4] investigated the long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation with step-like initial data, and obtained asymptotic formulas for the solution.Wang et al [5] studied the long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions at infinity by the nonlinear steepest descent method of Deift and Zhou, and found three asymptotic sectors in space-time plane, and gave asymptotic solutions for the three sectors.Finally, they also studied the modulational instability to reveal the criterion for the existence of modulated elliptic waves in the central region.In 2021, Bilman et al [6] studied the families of multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity, used the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, and computed the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions.And in 2022, Wang et al [7] applied the finite-gap integration approach and Whitham modulation theory to study the complete classification of solutions to the defocusing complex modified KdV equation with step-like initial condition.
In this paper, we will study a higher-order NLS-type equation with cubic and quintic nonlinear terms, that is, the Eckhaus-Kundu (EK) equation [8][9][10][11][12] where u(x, t) is a complex function about x and t, σ, δ are both real constants, δ 2 denotes the quintic nonlinear coefficient, 2δdenotes the nonlinear dispersion coefficient, the last term indicates the nonlinear term caused by the time-retarded induced Raman process.Equation (1) was given successively by Kundu [8], Calogero and Eckhaus [9] in their study on the integrability of the nonlinear Schrödinger equation (NLS), which has a wide range of applications in physics, optics and other fields.Such as: Clarkson [10] described the propagation of ultrashort femtosecond pulses in optical fibres in quantum field theory, investigated the problem of approaching the critical point of the strongly interacting many-body system of equation (1), and obtained its exact solution.Also, the exact solutions without and with current density were obtained.Johnson [11] showed how to extend the periodic wave modulation procedure in the classical water wave problem, through the process of deriving the higher order equations associated with the water waves of equation (1), the details of the instability of the Benjamin sidebands were investigated, these results can be used to study the change in form of soliton solutions near stable boundaries, which leads naturally to more general applications to 'soliton' solutions and inverse scattering methods.In nonlinear optics, Kodama [12] modelled a long-distance high-bit-rate transmission system, the propagation of optical solitons in single-mode fibres was discussed, and equation (1) was investigated with respect to the integrability of the perturbations, as well as considering the soliton as a stable fixed point of an infinite-dimensional mapping generated by a transmission system with periodic excitations.
Clarkson and Cosgrove in the literature [13] studied the Lax pair of equation (1).Wang et al [14] and Mendoza et al [15] studied the soliton solutions of equation (1).Xie et al [16] studied the rogue wave solutions of equation (1).Zha [17] studied the higher order rogue wave solutions of equation (1).A recent literature [18] studied the integrable discretization of equation (1), Tian et al gave x-discrete, t-discrete and a fully discrete form of the Eckhaus-Kundu equation based on the bilinear form, the single soliton and double soliton solutions of the derived discrete equation have been successfully constructed by the Hirota bilinear method.Cimpoiasua et al [19] studied the explicit invariant solution of equation (1) from the point of view of Lie symmetry analysis, the hyperbolic function type soliton solution of (1) was obtained.Luo and Fan [20] used the ¶ ¯-dressing method to construct the two-soliton solution and the N-soliton solution of equation (1).
In the study of nonlinear systems, it is great importance to consider the stability of the solutions.Pava J A [21] investigated the orbital stability of the dn periodic wave solutions of the focusing nonlinear Schrödinger equation where u = u(x, t) ä C, x, t ä R and the mKdV equation by applying the ideology for proving stability of solitary wave solutions proposed by Benjamin and Bona.Since then, the studies for orbital stability of dn periodic wave solutions in nonlinear systems has received a great deal of attention [22][23][24].Here we study the orbital stability of the periodic wave solution for the Eckhaus-Kundu equation (1).From reading the literatures, we understand that the research of orbital stability of dn periodic wave solutions about this type of equations has not appeared in the previous literatures.
It is worth noting that, the NLS and mKdV equations studied in the literature [21] have the highest number of their nonlinear terms only three times, and both are standard Hamilton systems.In contrast, the Eckhaus-Kundu equation (1) that we studied in this paper is not only a higher order NLS-type equation with cubic and quintic nonlinear terms, but also not a standard Hamilton system.Due to the fact that equation (1) has a higher order of nonlinear term than the equations studied in the literature [21], and the structure of the equation (1) is different from that of the equations studied in the literature [21], so that the methods used in this paper are different from those used in the literature [21].Since equation (1) is not a standard Hamilton system, the method proposed by Benjamin [25] and Bona [26,27] etc to prove the orbital stability of solitary wave solutions used in the literature [21] cannot be applied directly, and equation (1) has higher order term, these bring a lot of complexity to our calculation and proof.To do so, by constructing three conserved quantities, and using detailed spectral analysis and appropriate techniques, we conclude that the dn periodic wave solution of equation (1) is orbitally stable for small perturbations of the L 2 -norm.Then, as an extension on the process to prove the orbital stability of the periodic wave in equation (1), we prove the orbital stability of the solitary wave solution in equation (1).The proofs on the orbital stabilities of periodic and solitary wave solutions to the equation (1) in this paper complete and supplement the previous studies of Eckhaus-Kundu equation about stability.
The main points of this paper are: In section 2, the existence of dn periodic wave solution of equation ( 1) is proved by first integration and the properties of elliptic functions.In section 3, a detailed spectral analysis of the operator D was carried out.Using Floquet's theory [28], Laméʼs equation [29] and Wely's essential spectrum theorem [29], we obtained the spectral properties that the operator D qm ma 3 x 2 2 = - ¶ + + has three simple feature values, with zero as its second feature value and the rest of spectrum consisting of discrete multiple feature values.In section 4, by constructing three conserved quantities, and using the ideas from Benjamin [25] and Bona [26,27] for proving the stability of solitary wave solutions, it is shown that the orbital stability of dn periodic wave solution for Eckhaus-Kundu equation (1) with period L, and the orbital stability of the solitary wave solution for equation (1) at ω, v satisfy 4ω + v 2 < 0.

Existence of periodic traveling wave solution for equation (1)
In this section, we mainly study the existence of periodic traveling wave solution u x t e e a x vt e a , , 4 for the Eckhaus-Kundu equation (1), where a a x vt e a x vt x vt , , 5 where g(a From the fact that both the real and imaginary parts of (7) are equal to 0, we get substitute (10) into the imaginary part (9), take E 1 , E 2 such that its coefficient is equal to 0, then there are then we have that (9) is constant to 0, and (8) can be reduced to a m a qa 0, 12 where m = 2σ, q = − (4ω + v 2 )/8σ.Multiplying equation (12) by a x ¢( ), and integrating once with respect to ξ, we obtain that a(ξ) satisfies where M a is a non-zero integration constant, and M v 4 8 . Let F(a) = a 4 + 2qa 2 + M a , then the solutions of equation (13) depend on the roots of the polynomial F(a).Since we consider m < 0 in this paper, then we have Therefore, there are real symmetric roots ±A and ±B for −F(a).Without loss of generality, we assume that 0 From m < 0 and the nonnegativity of the left side of (14), we know that B < a < A, and A, B satisfy , then equation ( 14) can be changed into According to the properties of the elliptic functions [30], it results in and t(0) = 1, following the relationship t = a/A, we get the dn wave solution and then there is where Since the dn wave solution has a fundamental period 2K, i.e., dn u k ), where K = K(k) denotes the first type of complete elliptic integral, the fundamental period of the dn wave solution a(ξ) in (17) can be obtained as Write the above equation as a function relating only to B, it follows that in theorem 2.1, there is a unique B ≡ B(q) that allows the fundamental period L of the periodic traveling wave solution ) such that T L aq 0 = .
(1) There exists an interval W q 0 ( ) around q 0 , an interval V B 0 ( ) around B 0 , and the unique smooth function W q V B : ( ), such that for q W q 0 " Î ( ), B q Î P( ) and (2) The periodic traveling wave solution a a A q B q , , v , = w (• ( ) ( )) with fundamental fixed period L and satisfying equations ( 6) and ( 13) is related to can be obtained as smooth.
(3) W q 0 ( )can be chosen as From the conclusions in theorem 2.1, there is B q , 0 0 0 L = ( ) .In the following we prove that 0 from ( 22) and (23), it can be calculated that where )is a rigorous subtraction function with respect to B, and the replenishment modulus )is a strictly increasing function with respect to B q 0, Î -( ).Take the derivative of f k¢ It can be shown that f k¢ ( )is an increasing function with respect to k¢, and thus we have , from this, equation (26) holds, i.e. there is 0 . It follows from the knowledge of the implicit function theorem, there is a unique smooth function W q V B : ( ), so that for q W q 0 " Î ( ), q q , 0 L P = ( ( ) ) . W q 0 ( )is an interval surrounding q 0 , V B 0 ( ) is an interval surrounding B 0 .Therefore, conclusion (1) of theorem 2.1 is proved.
Again, since q 0 can be arbitrarily taken in the interval W mL , 2 2   2 , and depending on the uniqueness of the function Λ, it is possible to extend W q 0 ( ) . Conclusion (2) holds by using the smoothness of the function. , ) is a rigorous subtraction function, thus it follows by ( 22) that k(q) is a rigorous subtraction function on q.
Proof.By theorem 2.1, Λ is a rigorous subtraction function with respect to B. As for q W q 0 " Î ( ), , using the relationship k k 1 ( ), therefore under the relation , i.e.Π is a rigorous increasing function with respect to q. Taking the derivative of (22) with respect to q, we get This proves that q k q ( )  is a rigorous subtraction function, and corollary 2.1 is proved., Following the properties of elliptic functions [30] and corollary 2.1 we know, K k E k ( ) ( ) is a rigorous increasing function with respect to k, k(q) is a rigorous subtraction function with respect to q, so there is

Spectral analysis
For any operator u, v ä X = H 1 ([0, L]), there is a real inner product Let X * is the dual space of X, then X * = H −1 ([0, L]), and there exists a natural isomorphism I: X → X * which defines 〈Iu, v〉 = (u, v), where 〈•, • 〉 indicates the pairing between X and X * .
It follows from (31) and (32) that Next we study the spectral properties of linear operator D.
On the basis of Wely's spectral theorem [29], we have σ ess Next, by means of the perturbation theorem, Floquet's theory [28] and the Laméʼs equation [29] eigenvalue problem, we analyse the spectral characteristics of the operator D as follows.
We begin with an analysis of the periodic feature values problem of the operator D on [0, L].
By the theory of tight self-congruent operators, the spectrum of the operator D in (35) is a countably infinite set and λ n → + ∞ as n → + ∞ .By χ n we represent the eigenfunction corresponding to the eigenvalue λ n , hence a continuous differentiable function χ n with period L can be scaled up to the entire interval ).Using Floquet's theory [28], the semi periodic problem for problem (35) Since that problem (37) is also a self-conjugating problem, hence a set of eigenvalue columns {μ n |n = 0, 1, L }, n → + ∞ , μ n → + ∞ is obtained, and this eigenvalue column satisfies Let ζ n be the eigenfunction of the feature value μ n .As for ∀x, a function g is said to be semi periodic when it has nature g(x + L) = − g(x).Its period is 2L and its semi periodic is L. So, the period of the solution for the equation Therefore, the solution of equation (39) is stable on the intervals (λ 0 , μ 0 ), (μ 1 , λ 1 ), L , then the intervals (λ 0 , μ 0 ), (μ 1 , λ 1 ), L are stable intervals; the solution of equation (39) is unstable on the intervals (− ∞ , λ 0 ), (μ 0 , μ 1 ), (λ 1 , λ 2 ), L , then the intervals (− ∞ , λ 0 ), (μ 0 , μ 1 ), (λ 1 , λ 2 ), L are unstable intervals.The unstable interval (− ∞ , λ 0 ) is always available.Theorem 3.1.For the dn periodic wave solution a a A q B q , , . Then, the linear operator D qm ma 3 x ]) has three simple eigenvalues, i.e., 0 l , 1 l and 2 l , where 0 1 l = is the second eigenvalue, and whose corresponding eigenfunction a¢, the rest of spectrum consists of discrete multiple feature values.
Proof.By (40) we show that 0 1 2 l l = < .From equation (34) we have Da 0 ¢ = , it is known that the eigenvalue 0 corresponds to the eigenfunction a¢.It is easy to see that the eigenfunction a¢ has two zeros on L 0, [ ), the zero eigenvalues of the operator D can be judged to be where ρ and λ are related by the equation According to Floquet's theory [28] it is known that equation (41) has three instability intervals: r r ( ), where for i 0  , i r is the eigenvalue associated with the periodic problem (40), i q is the eigenvalue associated with the semi periodic problem (37).So, the first three eigenvalues 0 r , 1 r , 2 r are simple eigenvalues, the remaining eigenvalues 3 Here we give that k 4 Y are the eigenfunctions corresponding to the first three eigenvalues 0 r , 1 r , 2 r , and their period are K 2 .It is easy to see that on K 0, 2 [ ], 0 Y has no zeros, 2 Y has two zeros, and for k 0, 1 Combining with (42), we have Clearly r l  is an increasing function, then we have 0 . From this we can see, 0 1 l = is the second eigenvalue of the operator D, 0 0 l < .
Then, through the feature problem where the eigenvalue i q is related to i m by the equation 1 2 q = + .0 q , 1 q are the first two eigenvalues of the semi periodic problem (37).The corresponding eigenfunction of 0 q is x cn x dn x sm 0,
Where A is a self-associating operator and has only a single negative nature root 0 l , its corresponding feature vector f . Consider the dn wave solution a x ( ) defined in theorem 2.1, then Proof.
(1) Since the periodic traveling wave solution a x ( ) defined in theorem 2.1 is bounded, it is easy to see that , and . It is clear that the function column j f { } is delimited, and we still denote the subsequence of j The infimum of 0 g can be taken on a fetchable function 0 f ¹ .From lemma 3.1 and theorem 3.1, it follows that the operator D has the spectral characteristics in lemma 3.1.Thus there exists continuously differentiable, the derivative of equation ( 12) with respect to q, we have Following from lemma 3.1, there is 0 0 g  .In summary, 0 0 g = .
(2) Using the method of proof in (1), it can be shown that 0 g  .We then use the reduction to absurdity to prove that 0 g > .Suppose that 0 g = , then there exists a function Φ satisfying . With Lagrange's theorem, we have that α, λ, θ satisfy , it can be deduced that 0 a = .Also since Da 0 ¢ = , then , there is 0 l = .And then a V F = ¢, a¢ is orthogonal to a a 2 ¢, contradictions.Therefore, there is 0 g > .,

Orbital stability of the periodic wave solution for the Eckhaus-Kundu equation (1)
In this section we follow the main ideas of the classical approach proposed by Benjamin [25], Bona [26,27] et al, the orbital stability of the dn periodic wave solution U x t a e a , i for the Eckhaus-Kundu equation (1) is demonstrated by constructing three conservation functions E, Q 1 and Q 2 , where a(ξ) is given by theorem 2.1.
Since equation (1) has the symmetry of phase and translation, i.e., if u(x, t) is a solution of equation (1), then for any (y, θ) ä R × [0, 2π), e i θ u(x + y, t) is also a solution of equation (1), thus, the orbital stability is defined as follows: Definition 4.1.The orbit a y y R : , 0, 2 45 is stable under the action of the periodic flow produced by Eckhaus- Kundu equation (1), that is, for 0 e " > , there exists then for any t R Î , t q q = ( ) and y y t = ( ), the solution u x t , ( ) of the equation (1) with initial value u 0 satisfies u t e a y i 1 In the function space X = H 1 ([0, L]), we consider the initial value of equation ( 1) Let T 1 and T 2 be the one-parameter unitary group operators on X, defined as Derive (49), ( 50) with respect to s 1 , s 2 at s 1 = 0, s 2 = 0, we have According to the literature [32] it is known that, for any u H L 0, per s 0 Î ([ ]), s 0, equation (1) is globally adapted, i.e., for u(0) = u 0 (x), equation (1) has a unique solution u C R H L ; 0, ).In order to demonstrate the orbital stability of the periodic traveling wave solution, we build three new conservation quantities described below: where G u g s d s Easily verified, E(u), Q 1 (u) and Q 2 (u) are C 2 -functionals specified on the complex space X, and their first-order Fréchet derivatives are noted as , respectively.By calculation, we obtain It can be shown that E(u), Q 1 (u) and Q 2 (u) are invariant under the effect of T(• ), that is, for any s 1 , s 2 ä R, there are and for ∀t ä R, u(t) is the flow of (47), we have Next, we prove the orbital stability of the periodic wave solution for the Eckhaus-Kundu equation (1).
Theorem 4.1.Let L 0 > , v 0 > and v 4 0 2 w + < are arbitrary fixed.Allow for the smooth curve of the dn periodic traveling wave q mL a A q B q , 2 , , given by theorem 2.1.Then for ω, v such that ) , the orbit generated by U a x x = ( ) ˆ( ) in X is orbitally stable under the action of the periodic flow produced by equation (1).
Proof.For the classical approach proposed by Benjamin [25], Bona [26,27] and Weinstein [33], the solution a x ˆ( ) to equation (1) from theorem 2.1 is considered.Initially, with y L 0, )and t R Î , we define y u y t e a q m u y t e a , , , , By applying the method in [25][26][27] it can be seen that, on the interval In the following, we consider the perturbation term of the periodic wave a x ( ).Let From the minimum property of y y t t , , q q = ( ) ( ( ) ( )), we can obtain 0

| ( )
. Therefore, from the fact that a x ( ) satisfies equation (7) we get From (58) we obtain that the compatibility condition satisfied by , Using the translation and rotation invariants of E(u), Q u 1 ( ) and Q u 2 ( ) defined in (52), ( 53) and (54), expression (58), for each r 2  , the property of H L L L 0, 0, , and equations ( 7) and (11)satisfied by a x ( ), we obtain the variance of the conserved continuous functional u E u Q u vQ u > =|| || .In order not to lose generality, assuming that a 1 =

|| ||
, and let u a a a , from the definition of the operator D, and using the Cauchy-Schwarz inequality, we have 64 By the above conclusions we obtain where g x ax bx cx . For an arbitrarily small enough x, g(x) has the property g x 0 > ( ) .Let 0 e > , since u ϝ( ) holds.Finally, according to the mapping q mL a A q B q , 2 , , is continuous and the results derived from the previous analytical proofs, we obtain the conclusion that in space X H L 0, orbitally stable with small perturbations in the L 2 -parameter., In addition, the orbital stability of the solitary wave T t T vt a x ( ) can be discussed as follows.
Note 4.1: , the relationship can be obtained as A 2 + B 2 = − 2q.Further it follows from the properties of elliptic functions [30], where k is the modulus, k¢ is the replenishment modulus.So, if B → 0 + , there are k(B) → 1 − , A q 2  -.As in the elliptic functions, dn x sech x , 1 = ( ) ( ), from equation (17) we have This time, equation (17) loses periodicity in this limit to give a waveform with a single peak and 'infinite period', and the waveform is attenuated at infinity.Thus, we can obtain the bounded analytic solution where l = = − ω − v 2 /4 > 0, 2σ = m < 0. q, m are given by equation (12).This leads to the solitary wave solution u x t e e a e e l sech l x vt , , .6 9 of the equation (1).Based on the definitions of T 1 and T 2 in equations (49) and (50) we can write equation (69) as . Combining (5), ( 6) and (10), it can be verified that E a x Q a x vQ a x 0. 71 Define the operator of X → X * as = w w , which means that I −1 H ω,v is a bounded self-conjugate operator on X.The spectrum of H ω,v consists of the real number λ that make H ω,v − λI irreversible.From (51), ( 70) and (71) we calculate that and by (73) that Z is in the nucleus of the operator H ω,v .where Z is defined by (74), N is a finite subspace of X, such that P is a closed subspace of X, and there exists a positive constant ℓ 1 independent of u such that denote d″(ω, v) as the Hessian matrix of d(ω, v).We represent the number of positive eigenvalues of d″ by p(d″) and the number of negative eigenvalues of H ω,v by n(H ω,v ).
It follows from the literature [29] that local solution exists for the initial value problem (47), (48) of equation (1).From the analysis and discussion of the previous contents, it is clear that equation (1) has three conserved quantities satisfying (55), (56) and that the solitary wave of equation (1) satisfies (70).Also, we have defined the operator H ω,v .Therefore, according to the 'stability theorem' in the introduction to [34] where r 1 , r 2 , r 3 are real functions and r r X H R ,  ), l 0 > .Also due to Da x 0 ¢ = ( ) , thus, we obtain that the operator D has spectral properties: D has a unique simple negative eigenvalue, the nucleus of D is tensed by a x ¢( ), and the rest of the spectrum is positive and bounded away from zero.From the literature [35], for r H R < ¢ > =< Y > = , then there exists a positive real number 0 > ℓ , independent of r 1 , such that Dr r r , .8 1 , k 1 is an arbitrary real number.Then from (79), (80) we have  91), it follows that (84) holds.In this case, for any f(x) = e i ψ( x) (r 1 (x) + ir 2 (x)) ä X, chosen a 1 = 〈r 1 , Under the conditions of theorem 4.3, it follows from (68) that a x l sech lx ), so there are a a l , 2 ) .Substituting into (93) gives det d 4 0 .9 4 To sum up, it is clear that, under the conditions of theorem 4.3, we have p(d″) = n(H ω,v ) = 1, that is, if σ < 0 and ω, v satisfy 4ω + v 2 < 0, the solitary wave T t T vt a x

Conclusion
In this paper we focus on the orbital stability of the periodic wave solution of the Eckhaus-Kundu equation (1).Since equation (1) is not a standard Hamilton system, the method suggested by M. Grillakis [34,35] et al for proving the orbital stability of solitary wave solutions cannot be applied directly, and this equation has two higher order nonlinear terms.To this end, we construct three conserved quantities and use a special technique to overcome the above difficulties of equation (1) and prove that the dn periodic wave solution of equation ( 1) is orbitally stable under small perturbations of the L 2 -parameter.As an extension of the proof for the above results, we also prove the orbital stability of the solitary wave in equation (1).This discussions about orbital stabilities of periodic and solitary wave solutions to the Eckhaus-Kundu equation (1) in this paper complete and supplement previous studies of Eckhaus-Kundu equation on stability.
Thus by theorem 3.2 (2), it follows that

Hypothesis 4 . 1 .
(Spectral decomposition of H v , w ) Space X can be divided into direct sum

) 1 l 1 
then the operator D can be written as D l It follows from the spectral properties of the operator D in theorem 3.1, 0 = is the second eigenvalue of D, and D has only one negative eigenvalue 0 l , whose corresponding eigenfunction is 0 Y , i.e. there is D 0 0 0 l Y = Y .From (68) in Note 4.1, as x ¥ | | , a 0 2  , it follows that M x 0 ( ) .According to Wely's essential spectrum theorem[29], we have D DΨ 0 = λΨ 0 , then f(x) can be uniquely expressed as In summary, Hypothesis 4.1 holds.Next prove that p(d″) = n(H ω,v ) = 1, i.e. prove det d ˆ) of the Eckhaus-Kundu equation (1) is orbitally stable on space X H R .
1, L ; the period of the solution for equation (39) is 2L if and only if δ = μ n , n = 0, 1, L .If the solution of equation (39) is bounded, then such a solution is said to be stable, otherwise it is said to be unstable.The zeros of χ n and ζ n on [0, L] can be obtained by analysis: χ 0 has no zeros; χ 2n+1 and χ 2n+2 have 2n + 2 zeros; ζ 2n and ζ 2n+1 have 2n + 1 zeros.It follows from the perturbation theorem that (36) and (38) are interleaved, i.e. there is (1)]heorem 4.1 in[34], we can obtain the following orbital stability theorem for the solitary wave of equation(1).According to theorem 4.2, the main conclusion of equation (1) for the orbital stability of solitary wave can be obtained.Proof.From the previous discussion, it can be seen that equation (1) has solitary wave T t T vt a x