Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term

In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: 0.1{iut+uxx=αuv+γ|u|2u+δ|u|4u,vt+β|u|x2=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$\end{document} We first obtain explicit exact solitary waves for Eqs. (0.1). Second, by applying the extended version of the classical orbital stability theory presented by Grillakis et al., the approach proposed by Bona et al., and spectral analysis, we obtain general results to judge orbital stability of solitary waves. We finally discuss the explicit expression of det(d″)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\det (d^{\prime \prime })$\end{document} in three cases and provide specific orbital stability results for solitary waves. Especially, we can get the results obtained by Guo and Chen with parameters α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha =1$\end{document}, β=−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta =-1$\end{document}, and δ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta =0$\end{document}. Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with γ=δ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma =\delta =0$\end{document} and the orbital instability results for the nonlinear Schrödinger equation with β=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta =0$\end{document}.


Introduction
Long-short (LS) wave interaction equations have been proposed for many physical problems, such as internal, Rossby, and plasma waves. Kuznetsov et al. [1] proposed some generalized LS-type coupled equations. In this paper, we investigate one type of the generalized LS wave resonance equations with cubic-quintic strong nonlinear term ⎧ ⎨ ⎩ iu t + λu xx = αuv + γ |u| 2 u + δ|u| 4 u, x ∈ R, v t + β|u| 2 x = 0, x ∈ R. (1.1) When v = 0, Eqs. (1.1) reduce to the nonlinear Scrödinger equation describing electromagnetic wave propagation in nonlinear isotropic dielectrics, for example, in an isotropic plasma. In this case, u denotes the complex amplitude of the electric field, and γ |u| 2 u + δ|u| 4 u is the nonlinear addition to the refraction index. However, for many problems, accounting for a finite time of medium relaxation is critical. Thus, for electromagnetic radiation propagation in an isotropic plasma, the nonlinear frequency shift is caused by density modulation under the action of a powerful wave, and the coupled equation was proposed.
In 2005, Shang [2] studied the explicit and exact special solutions of Eqs. (1.1), where α, β, γ , λ, and δ are all real constants with λαβ = 0. The quintic term δ|u| 4 u in the first equation of (1.1) describes the strong nonlinear self-interaction in the high-frequency subsystem, which corresponds to a self-focusing effect in plasma physics. Obviously, if γ = δ = 0, then Eqs. (1.1) reduce to the classical LS wave equations (1.2) Equations (1.2) were first derived by Djordjevic and Redekopp [3] to describe the resonance interaction between long and short waves. In Eqs. (1.2), u is a complex-valued function and denotes the envelope of the short wave, and v is a real-valued function and denotes the amplitude of the long wave. As highlighted in [3], the physical significance of Eqs. (1.2) is that the dispersion of the short wave is balanced by the nonlinear interaction of the long and short waves, whereas the evolution of the long wave is driven by the self-interaction of the short wave. These equations also appear in an analysis of internal waves [4] and Rossby waves. In plasma physics, similar equations can be used to describe the resonance between high-frequency electron plasma oscillations and associated lowfrequency ion density perturbations [5]. Ma [6] found that Eqs. (1.2) can be rewritten in Lax's formulation, and the Cauchy problem of Eqs. (1.1) can be solved by the inverse scattering method. Adapting the method developed by Bona and Weinstein, Laurencot [7] confirmed that the solitary wave solution of (1.2) was stable. Moreover, if δ = 0, then Eqs. (1.1) reduce to the LS wave resonance equations where α, β, γ , λ ∈ R with λαβ = 0. Equations (1.3) were a particular case of the equations proposed by Benney [8]. In that study, Benney provided a general theory for deriving nonlinear partial differential equations that allow both long and short wave solutions. By an appropriate change of both independent and dependent variables, we can take λ = 1, α = 1 and β = -1 to obtain (1.4) System (1.4) arises in the study of surface waves with both gravity and capillary modes being present [9] and in plasma physics [10]. We can say that Eqs. The well-posedness of the local solution or/and global solution for the initial value problem and periodic initial value problem of system (1.4) and its extensions have been investigated by several authors. Among these, we refer the reader to [11][12][13][14][15]. The existence of global attractors and approximation inertial manifolds have been studied by many researchers [16][17][18][19][20][21][22][23][24]. Guo and Chen [25] studied the orbital stability of solitary waves of (1.4) by applying the abstract results of Grillakis et al. [26,27]. Unfortunately, the conditions that ensure the orbital stability of solitary waves were incorrect because of incorrectness of d cc (ω, c) and consequently of det(d ) (see p. 893 of [25]). Based on the qualitative theory and bifurcation theory of planar dynamical systems, a series of explicit and exact solutions of solitary waves for Eqs. (1.1) were obtained by seeking the homoclinic and heteroclinic orbits for a class of Liénard equations [2]. An interesting problem is whether the solitary waves of the generalized LS wave equations with a cubic-quintic strong nonlinear term (1.1) are orbitally stable or instable. However, till date, to the best of our knowledge, no research has been conducted on the orbital stability of the solitary waves of the generalized LS wave equations having a cubic-quintic strong nonlinear term (1.1).
In this paper, we consider the existence and orbital stability of solitary waves for the generalized LS wave equations (1.1). We focus on solutions for (1.1) of the form where ω, c ∈ R, ξ = xct, φ ω,c , ψ ω,c : R → R are smooth functions, and φ ω,c (ξ ), ψ ω,c (ξ ) → 0 as |ξ | → ∞. It is worth pointing out that Eqs. (1.1) contain two nonlinear terms. Our results contain the orbital stability of solitary waves for the classical LS wave equation with γ = δ = 0, the orbital instability results for the nonlinear Schrödinger equation with β = 0, and the orbital stability of solitary wave for LS wave equation with one nonlinear term.
Because here the stability refers to perturbations of the solitary wave profile itself, a study for the initial value problem of (1.1) is necessary. Similarly to Theorem 1.2 in [14], by using Banach's fixed point theorem and employing some smoothing-effect estimates, after slightly modifying the proof of [14], we obtain the well-posedness of the initial value problem of (1.1).

Theorem 1 For any
The orbital stability of solitary waves is defined as follows.
By applying the extended version of the general theory of orbital stability presented by Grillakis et al. [26], the lines of the stability theorem in the introduction of [27] or Theorem 4.1 in [27], the approach in [28], and detailed spectral analysis, we obtain the following abstract stability results of solitary waves (1.5) for Eqs. (1.1).

Theorem 2 Assume that (1.1) has a family of solitary waves that belong to H
. Moreover, suppose that φ ω,c has one simple zero and decays rapidly to zero at ±∞. Then the solitary wave The remainder of this paper is structured as follows: For convenience, we first introduce the existence of solitary waves for the generalized LS wave equations (1.1). Then, in Theorem 4, we present the spectral analysis of some certain self-adjoint operators necessary to obtain our stability result and state the stability results. Finally, we prove the stability results under three conditions.

Exact solitary waves of the generalized LS wave equations with cubic-quintic nonlinearity term
For convenience, in this section, we consider the solitary wave solutions of the following generalized LS wave resonance equations with a cubic-quintic strong nonlinear selfinteraction term: with real α, β, γ , δ. Assume that Eqs. (2.1) have solutions of the form where ω, c are real numbers, and a, φ ω,c , and ψ ω,c are real functions. We set ξ = xct and assume φ ω, Without loss of generality, we assume that D = 0, and then a(ξ ) = c 2 ξ . By collecting (2.6) and a (ξ ) = c 2 , Eq. (2.4) becomes Multiplying by 2φ ω,c both sides of Eq. (2.7) and integrating from -∞ to ξ , it follows that is the form of (3.25b) in [29]. Then, according to [29] (also see [30]), there exists a solitary wave of the form Therefore we have the following lemma. (2.10) Especially, when d 4 = 0, we obtain the following solution of Eq. (2.8): where φ ω,c (ξ ) and ψ ω,c (ξ ) are given by (2.10) and (2.6), respectively.

Verification of conditions that enable Eqs. (1.1) and its solitary waves to satisfy the abstract stability theory
In this section, we prove that Eqs. (1.1) are a Hamiltonian system and satisfy the conditions of the general orbital stability theory proposed by Grillakis et al. [26,27] for some parameters.
In [25] the authors rewrite (1.4) in terms of real and imaginary parts and reduce Eqs. (1.4) to Eqs. (3.1). Then they define the function space wherein they work on and develop their analysis. In this paper, we define the function space wherein we work on and develop our analysis directly starting from Eqs. (1.1).
Let U = (u, v) T . The function space we will work on is defined by X = H 1 complex (R) × L 2 (R). Let the inner product of X be where ·, · denotes the dual product between X and X * ,
Next From Theorem 2 we obtain the following main results regarding the orbital stability of solitary waves for Eqs. (1.1).

Orbital stability of solitary waves for Eqs. (1.1) in three cases
In this section, we verify that p(d ) = 1 under the conditions of Theorem 2 and provide a detailed proof of Theorem 4.
Furthermore, when β = 0, according to the instability theory [27] (n(H ω,c )-p(d ) is odd), we can obtain the following result by the same process as that detailed in Sect. 3 and Case (b).