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 $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ 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 $\det (d^{\prime \prime })$ 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 $\alpha =1$ , $\beta =-1$ , and $\delta =0$ . Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with $\gamma =\delta =0$ and the orbital instability results for the nonlinear Schrödinger equation with $\beta =0$ .

中文翻译：

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三次三次强非线性项广义长短波共振方程孤波的轨道稳定性

在本文中，我们针对以下哈密顿量形式的广义长短波共振方程，研究了孤波的轨道稳定性：0.1 $$ \ textstyle \ begin {cases} iu_ {t} + u _ {{xx}} = \ alpha uv + \ gamma \ vert u \ vert ^ {2} u + \ delta \ vert u \ vert ^ {4} u，\\ v_ {t} + \ beta \ vert u \ vert ^ {2} _ {x} = 0 。\ end {cases} $$首先，我们为方程式获得明确的精确孤波。（0.1）。其次，通过应用Grillakis等人提出的经典轨道稳定性理论的扩展版本，Bona等人提出的方法以及频谱分析，我们获得了判断孤波轨道稳定性的一般结果。最后，我们讨论了在三种情况下$ \ det（d ^ {\ prime \ prime}）$的显式表达式，并为孤波提供了特定的轨道稳定性结果。特别，我们可以得到Guo和Chen使用参数$ \ alpha = 1 $，$ \ beta = -1 $和$ \ delta = 0 $获得的结果。此外，我们可以获得$ \ gamma = \ delta = 0 $的经典长短波方程的孤波的轨道稳定性，以及$ \ beta = 0 $的非线性Schrödinger方程的轨道不稳定性结果。