Persistence of Kink and Periodic Waves to Singularly Perturbed Two-Component Drinfel’d–Sokolov–Wilson System

This paper concerns the persistence of kink and periodic waves to singularly perturbed two-component Drinfel’d-Sokolov-Wilson system. Geometric singular perturbation theory is first employed to reduce the high-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we then are able to find the sufficient conditions about the wave speed to guarantee the existence of heteroclinic orbit and periodic orbits, which indicates the existence of kink and periodic waves. Furthermore, we also show that the limit wave speed c0(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0(k)$$\end{document} is increasing.


Introduction
As we know, partial differential equations (PDEs) provide a good way to model the phenomena in real world and their studies have significant applications in many fields such as mathematics, physics, engineering, and so on.A lot of efficient and effective numerical and analytical methods have been developed to study the solutions and their dynamical behaviors.As we know, the classical two-component Drinfel'd-Sokolov-Wilson (DSW) system [1][2][3] 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:980-995 with parameters p, q, a and b, is an important water wave model, which is used to describe the nonlinear surface gravity waves propagating over horizontal seabed.System (1) was originally introduced by Drinfel'd and Sokolov [1,2] and Wilson [3] from the shallow water wave models.Later, Hirota et al. [4] also derived system (1) from the Kadomtsev-Petviashvili hierarchy.Due to the great applications in physical fields, there have been considerate work concerning the solutions and their dynamics of system (1) and its variants.For example, Hirota et al. [4] provided the soliton structure of system (1).Yao and Li [5] constructed some exact solutions of system (1) through a direct algebra method.Likewise, Liu and Liu [6] presented its four kinds of exact solutions algebraically and revealed their relations.Fan [7] devised an algebraic method to uniformly construct a series of exact solutions of system (1).By improving generalized Jacobi elliptic function method, Yao [8] found some traveling wave solutions of system (1).Doubly periodic wave solutions of system (1) were also constructed through the Adomian decomposition method [9] and an improved F-expansion method [10].The bifurcation method [11] and the Expfunction method [12] were also employed to derive exact solutions of system (1).Additionally, some methods including the modified kudryashov method [13], the first integral method [14] and the bifurcation method [15,16] were extended to find abundant exact solutions of fractional DSW system.However, we find that there is little work concerning singularly perturbed DSW system.In fact, in modelling real world problems, such as the shallow water waves in nonlinear dissipative media [17] and dispersive media [18], some relatively weak influences due to uncertainty or perturbation are unavoidable.Therefore, one generally should include certain type of small perturbation to obtain a more realistic model and to better understand its dynamics.Recently, the singularly perturbed models widely appear in modeling the problems across many areas of the natural sciences, and has attracted more and more interest [19][20][21][22][23][24][25][26][27][28][29][30].Ogawa [19] investigated the persistence of solitary waves and periodic waves of the perturbed KdV equation and Yan et al. [20] further proved the results for a perturbed generalized KdV equation.Further, Chen et al. [22] also considered the persistence of kink and periodic waves for a perturbed defocusing mKdV equation.Chen et al. [21] also obtained the persistence of solitary waves and periodic waves for the perturbed generalized BBM equation.Among most of these work [19-21, 23, 24, 26, 27], the authors focused on solitary waves and periodic waves, and little [22,25] concerned kink waves, especially for the two-component systems.As a matter of fact, kink waves have been found in many important integrable models [31][32][33][34][35][36] including negative-order KdV equation and Camassa-Holm equations, and are believed to have many significant applications in fluid mechanics, nonlinear optics, classical and quantum fields theories etc.Therefore, it is of interest to check whether these kink waves persist under perturbation.
In this paper, we intend to examine the dynamics of the following singularly perturbed two-component DSW system (1) w t + puu x = 0, u t + awu x + buw x + qu xxx = 0, where the parameters p < 0, a > 0, b > 0, q > 0 , and small  > 0 standing for the perturbation parameter.In system (2), u xx and u xxxx represent the backward diffusion and dissipation terms, respectively.At first glance, one might expect that system (2) should have similar solutions with system (1).As indicated in [19], the answer to this question is significant not only mathematically but also in the application point of view, since in some physical circumstances, for example the Reynolds number is large or surface tension is small, correspond to the case when is small.Besides, it also helps understand the role of dispersion, dissipation, and instability in nonlinear wave systems.Our aim is to show the persistence of kink and periodic waves with singular perturbation under suitable conditions.To be specific, we first reduce the corresponding high-dimensional system to the perturbed planar system by geometric singular perturbation theory (GSPT).Then we are able to study the wave speed in detail by perturbation analysis and Abelian integrals theory, from which we find the sufficient conditions to guarantee the existence of heteroclinic orbit and periodic orbits, which indicates the existence of kink and periodic waves.Furthermore, we also show that the limit wave speed c 0 (k) is increasing for k ∈ [− 1 4q , 0).

The reduction of traveling wave system on the slow manifold
In this section, we derive the reduction of traveling wave system corresponding to (2) on the slow manifold by exploiting GSPT.The transformations with c > 0 , convert system (2) into the following system The first equation of (3) yields with constant g 1 .Substituting it into the other equation of (3) yields , where 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:980-995 where dot indicates the derivative with respect to .System (5) indicates the following singularly perturbed system which is called the slow system, and has three equilibrium points (0, 0, 0), (−1, 0, 0) and (1, 0, 0).
Introducing the transformation = , we obtain the following equivalent fast sys- tem corresponding to system (6) Letting = 0 in system (6), we obtain the critical manifold Note that the linearization of the fast system (7) restricted on C 0 is given by the fol- lowing matrix Obviously, the matrix M has three eigenvalues 1 = 0, 2 = 0 and 3 = −q √ ag 1 − c , and therefore C 0 is normally hyperbolic.According to GSPT, for  > 0 sufficiently small, three exists a two-dimensional submanifold C of R 3 within the Hausdorff dis- tance of C 0 , which is invariant under the flow of system (6).
One can write where Z( , y, ) depends smoothly on , y, , and satisfies Z( , y, 0) = 0 .We can expand Z( , y, ) in as follows (5) By substituting it into the slow system (6), we can get Equating the coefficients of , we can obtain Therefore, the dynamics on the slow manifold C for system ( 6) is determined by It is easy to check that system (10) with = 0 is a Hamiltonian system with the Hamiltonian function and its phase portrait is given in Fig. 1.In addition, we have H(0, 0) = 0 and H(±1, 0) = − 1 4q .Then we can parameterize the traveling waves of system (10) with = 0 through the curves H( , y) = k with parameter k, and display the existence result of traveling wave solutions of system (2) as in the following theorem.
Theorem 2.1 For the perturbed two-component DSW system (2), we have the following results.
Remark 1 It is well-known that system (10) with = 0 has heteroclinic orbit and a family of periodic orbits, which correspond to kink and periodic waves of system (1) [11], and it is natural to ask whether these heteroclinic orbit and periodic orbits will break up or persist under perturbation in system (2).Theorem 2.1 provides the sufficient conditions about the wave speed to guarantee the persistence of heteroclinic orbit and periodic orbits of system (2), which indicates the existence of kink and periodic waves to system (2).
Journal of Nonlinear Mathematical Physics (2023) 30:980-995 −∞ <  < 0 and y(0) = y 1 , where y 1 is the y coordinate corresponding to the inter- section point of the unstable manifold of (−1, 0) and the y axis (see Fig. 2b).The later is defined similarly.Following the similar procedure, we also deduce that the limit speed c 0 (k) satisfies (15), where 0 is a solution of system (10) with = 0 and the integration is performed on the curve H = − 1 4q .

Analysis by the Abelian Integral Theory
In this section, we first express the limit speed c 0 (h) in the form of Abelian integrals and then study its properties.Furthermore, we will prove the Theorem 2.1.Assume that ( ) is the solution of system (10) with = 0 , and Q and R are defined by where the integrals are performed along the orbits of system (10).
Introducing a new variable h = 4qk , now from ( 15), we can treat c 0 (k) as a function of h Now it is time to analyze Q and R in detail.Suppose that ± (h) and ± (h) are four roots of 4 − 2 2 − h = 0 , where −1 ≤ h < 0 , satisfying 0 ≤ (h) ≤ (h) .There- fore, we can express Q and R as where E( ) = √ 4 − 2 2 − h.For convenience, we introduce the following integrals: which satisfy by direct calculus.Now we can rewrite Q and R as To study the monotonicity of c 0 (h) , we turn to Z(h) = Q R , and present its properties in Proposition 4.1.

Moreover
To prove Proposition 4.1, we need the following lemmas.
, and , and lim 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:980-995 and it follows that by integration by parts and (18).Hence, we have Similarly, we have and it follows that, J 4 and J 6 can be obtained in a similar way.Here we omit them.◻ From ( 19) and ( 20), we see that Q can be expressed by J ′ 0 and J ′ 2 , which can be inversely expressed by J 0 and J 2 in Lemma 4.3.
Proof This lemma follows from the first two equations in (20) by direct calculation.◻ Lemma 4. 4 We have .

Conclusion
Through perturbation analysis and Abelian integrals theory, we derive the sufficient conditions about the wave speed to guarantee the existence of heteroclinic orbit and periodic orbits, which indicates the existence of kink and periodic waves.Besides, we also prove the monotonicity of the limit wave speed c 0 (k) .As we know, the wave phenomena, such as the discovery of solitary waves, kink and periodic waves, play an important role in fluid dynamics, plasma and elastic media.We believe that the waves found under small perturbation in a more realistic model will help facilitate the wave dynamics, and it potentially provides a way to analyze the propagation of the nonlinear waves.

Lemma 4 . 1 Lemma 4 . 2
We have Proof Through direct calculus, we have and J 2 (−1) J 0 (−1) = 1 5 follows.In addition, by the squeeze theorem, we easily get ◻ We have Proof Differentiating both sides of E 2 = 4 − 2 2 − h with respect to yields

ag 1
−c 0 (k)(ag 1 −c 0 (k)) > 0 , by the implicit func- tion theorem, there exist a unique smooth function c h ( ) = c( , h) for h ∈ [−1, 0] and ∈ (0, 0 ) so that The existence of c( , h) indicates the first statement in Theorem 2.1, since k = h 4q .While the second statement in Theorem 2.1 follows from Proposition 4.1, and the third statement holds obviously.

Fig. 3
Fig. 3 Illustration of contradiction and it follows that which indicates the statement.