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

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.

Stability of solitary waves on deep water with constant vorticity

&lt;p&gt;Waves on deep water with constant vorticity propagating in the direction of the shear are known to be weakly dispersive in the long wave limit. Weakly-nonlinear evolution of such waves can be described by the Benjamin-Ono equation, which is integrable and has stable soliton solutions. In the present study we investigate behaviour of finite-amplitude counterparts of Benjamin-Ono solitons by modelling their dynamics within exact equations of motion (Euler equations). Due to the solitons having a near-Lorentzian shape with slowly decaying tails, we need to approach them by examining periodic waves, whose crests, indeed, become more and more localised as the period increases. We perform a parameter space study and analyse how stability of very long waves depends on their amplitude and period. We show that large-amplitude solitary waves are unstable.&lt;br&gt;This research was supported by RFBR (grant No. 16-05-00839) and by the President of Russian Federation (grant No. MK-2041.2017.5). Numerical experiments were supported by RSF grant No. 14-17-00667, data processing was supported by RSF grant No. 15-17-20009.&lt;/p&gt;

On the orbital stability of traveling waves that behave such as particles

Properties that solitary waves share with particles have contributed significantly to the development of new theories and technological advances in different areas of knowledge. In this sense, the study of orbital stability of solitary waves is key in solitary wave dynamics. Although the definition of orbital stability is relatively simple, the mathematical analysis required to verify it is quite complex. However, the theory of Grillakis, Shatah and Strauss provides us with a very useful criterion to verify orbital stability. In this work, we present their theory and apply it to analyse the orbital stability of Generalized Korteweg-de Vries equation, Compressible fluid equation, and one-dimensional Benney-Luke equation. For the first two equations, the criterion guaranteed the orbital stability of the solitary waves. For the third one, it was guaranteed only for certain ranges of its parameters.

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: $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ { 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 .