scholarly journals Orbital stability of solitary waves to the coupled compound KdV and MKdV equations with two components

2020 ◽  
Vol 5 (4) ◽  
pp. 3298-3320
Author(s):  
Xiaoxiao Zheng ◽  
◽  
Jie Xin ◽  
Yongyi Gu ◽  
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Stability Of Solitary Waves

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

Analytic and numerical nonlinear stability/instability of solitons for a Kawahara-like model

2016 ◽  
Vol 14 (04) ◽  
pp. 479-501
Author(s):  
José R. Quintero ◽  
Juan Carlos Muñoz
Keyword(s):  
Wave Velocity ◽  
Solitary Waves ◽  
Nonlinear Stability ◽  
Water Waves ◽  
Small Amplitude ◽  
Orbital Stability ◽  
Type Equation ◽  
Unstable Behavior ◽  
Stability Of Solitary Waves ◽  
Least Energy

We study orbital stability of solitary waves of least energy for a nonlinear Kawahara-type equation (Benney–Luke–Paumond) that models long water waves with small amplitude, from the analytic and numerical viewpoint. We use a second-order spectral scheme to approximate these solutions and illustrate their unstable behavior within a certain regime of wave velocity.

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

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoxiao Zheng ◽  
Huafei Di ◽  
Xiaoming Peng
Keyword(s):  
Solitary Waves ◽  
Nonlinear Term ◽  
Orbital Stability ◽  
Short Wave ◽  
Extended Version ◽  
Gamma Delta ◽  
Wave Resonance ◽  
Stability Of Solitary Waves ◽  
Orbital Instability ◽  
Stability Results

Abstract In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ { i u t + u x x = α u v + γ | u | 2 u + δ | u | 4 u , v t + β | u | x 2 = 0 . 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 $\det (d^{\prime \prime })$ det ( d ″ ) 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 $\alpha =1$ α = 1 , $\beta =-1$ β = − 1 , and $\delta =0$ δ = 0 . Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with $\gamma =\delta =0$ γ = δ = 0 and the orbital instability results for the nonlinear Schrödinger equation with $\beta =0$ β = 0 .

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