Orbital Stability Of Solitary Waves Research Articles

Orbital stability of solitary waves for a two-component Novikov system

Orbital stability of solitary waves for a two-component Novikov system

Orbital stability of solitary waves on a ferrofluid jet

We prove the orbital stability of small-amplitude axisymmetric solitary waves on the surface of an incompressible, inviscid ferrofluid jet. The ferrofluid surrounds a current-carrying rod and is subject to the azimuthal magnetic field generated by the rod. We show that under appropriate assumptions on the magnitude of the magnetic intensity in the ferrofluid, both the trivial flow and the solitary waves with strong surface tension are conditionally orbitally stable, while the conditional orbital stability of solitary waves with near-critical surface tension can be deduced from properties of the corresponding dispersive PDE model equation. The arguments are based on the recent orbital stability results for internal waves by Chen and Walsh (2022) and an improved version of the Grillakis–Shatah–Strauss method introduced by Varholm et al. (2020).

Orbital stability of solitary waves and a Liouville-type property to the cubic Camassa–Holm-type equation

We classify the existence and non-existence of a class of localized solitary waves for the cubic Camassa–Holm-type equation which arises as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves. In addition to those peaked solitary-wave solutions, we show by the phase portrait method that there are smooth and cusped solitary waves according to the wave speed and coefficients of nonlocal linear and nonlinear dispersions. We then prove that the smooth solitary-wave solutions are orbitally stable in the energy space under small perturbation. Finally, we establish a Liouville-type property of the solution.

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.

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: 0.1{iut+uxx=αuv+γ|u|2u+δ|u|4u,vt+β|u|x2=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\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^{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.

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

This paper is to study the following coupled version of compound KdV and MKdV equations with two components $ \left\{ \begin{array}{l} {u_t} + \alpha v{v_x} + \beta {u^2}{u_x} + {u_{xxx}} + \lambda u{u_x} = 0,\beta \gt 0,\\ {v_t} + \alpha {(uv)_x} + 2v{v_x} = 0, \end{array} \right. $ which clearly has Hamiltonian form. The orbital stability and instability of solitary waves with nonzero asymptotic value have been few studied. In this paper, we mainly consider the orbital stability and instability of solitary waves with zero or nonzero asymptotic value for this equations. Precisely, we first obtain two explicitly exact solitary waves with zero asymptotic value and four explicitly exact solitary waves with nonzero asymptotic value. Secondly, we conclude some results on the orbital stability of solitary waves with zero or nonzero asymptotic value. To this aim, in order to overcome the difficulty in studying orbital stability of solitary waves with nonzero asymptotic value, we use a translation transformation to transfer this problem into solitary waves with zero asymptotic value for a reduced nonlinear coupled equations. Then by applying the classical orbital stability theory presented by Grillakis et al. and Bona et al., we obtain the orbital stability and instability of solitary waves with zero asymptotic value for the new equations. We finally derive some results on orbital stability of solitary waves with zero or nonzero asymptotic value. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition $v = 0$, called the compound KdV and MKdV equation, which have been studied by Zhang et al.

Orbital Stability of Solitary Waves for Generalized Derivative Nonlinear Schrödinger Equations in the Endpoint Case

We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $\sigma\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{\omega,c}(t,x)=\Phi_{\omega,c}(x)e^{i\omega t+\frac{ic}2x-\frac i{2\sigma+2}\int_0^x\Phi_{\omega,c}(y)^{2\sigma}dy},$$ with $(\omega,c)$ satisfying $\omega>c^2/4$, or $\omega=c^2/4$ and $c>0$. The stability theory in the frequency region $\omega>c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $\omega=c^2/4$ and $c>0$.

Orbital stability of solitary waves for derivative nonlinear Schrödinger equation

In this paper, we show the orbital stability of solitons arising in the cubic derivative nonlinear Schrodinger equations. We consider the zero mass case that is not covered by earlier works. As this case enjoys L2 scaling invariance, we expect orbital stability (up to scaling symmetry) in addition to spatial and phase translations. We also show a self-similar type blow up criterion of solutions with the critical mass 4π.

Orbital stability of solitary waves for generalized Boussinesq equation with two nonlinear terms

This paper investigates the orbital stability and instability of solitary waves for the generalized Boussinesq equation with two nonlinear terms. Firstly, according to the theory of Grillakis–Shatah–Strauss orbital stability, we present the general results to judge orbital stability of the solitary waves. Further, we deduce the explicit expression of discrimination d′′(c) to judge the stability of the two solitary waves, and give the stable wave speed interval. Moreover, we analyze the influence of the interaction between two nonlinear terms on the stable wave speed interval, and give the maximal stable range for the wave speed. Finally, some conclusions are given in this paper.

Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction

We study the existence and stability of standing waves solutions of a three-coupled nonlinear Schrödinger system related to the Raman amplification in a plasma. By means of the concentration-compactness method, we provide a characterization of the standing waves solutions as minimizers of an energy functional subject to three independent L2 mass constraints. As a consequence, we establish existence and orbital stability of solitary waves.

On the stability of the solitary waves to the (generalized) Kawahara equation

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space H2(R) of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [4] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [9]. The second family consists of even traveling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.

Ground state solitary waves for the planar Schrödinger-Poisson system

In this paper, we investigate the planar Schrödinger–Poisson System. Based on fixed point argument, Riesz’s rearrangement, Hardy–Littlewood–Sobolev inequality and critical point theory, we prove the existence and symmetry properties of ground state solitary waves. In addition to their existence, we also obtain the orbital stability of solitary waves.

On the linearized log-KdV equation

The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the linearized log-KdV equation at the Gaussian solitary waves. By using the decomposition of solutions in the energy space in terms of Hermite functions, we show that the time evolution is related to a Jacobi difference operator with a limit circle at infinity. This exact reduction allows us to characterize both spectral and linear orbital stability of solitary waves. We also introduce a convolution representation of solutions to the log-KdV equation with the Gaussian weight and show that the time evolution in such a weighted space is dissipative with the exponential rate of decay.

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations urn:x-wiley:mma:media:mma4187:mma4187-math-0001 where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

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

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.

Orbital stability of solitary waves of compound KdV-type equation

In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of \({u_t} + a{u^p}ux + b{u^{2p}}ux + {u_{xxx}} = 0\left( {b \geqslant 0,p{\text{ > }}0} \right)\). Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term \(b{u^{2p}}ux\), but also the nonlinear term \(a{u^p}ux\). For the case of b > 0 and 0 0, while that unstable when a 0.

Orbital stability and dynamical behaviors of solitary waves for the Camassa–Holm equation with quartic nonlinearity

In this paper we prove that the Camassa–Holm equation with quartic nonlinearity is non-integrable via the Painlevé method. The orbital stability of solitary waves for this equation is investigated by constructing a functional extremum problem. This result demonstrates that the resulting solitary wave is unstable when its speed lies in the narrow region of the critical value that connects with the bifurcation condition. In contrast when the speed surpasses the narrow region, the solitary wave is stable. In addition, the stable solitary wave turns into a chaotic state when is driven externally. If a damping term controller is added to the perturbed equation, the solitary wave can also propagate stably under a certain condition. Finally our numerical results show that the perturbed equation is not well controlled when a certain resonant-frequency occurs and is well controlled with a smaller wave speed as well as a higher nonlinear convection.

Orbital Stability of Solitary Waves for Generalized Symmetric Regularized-Long-Wave Equations with Two Nonlinear Terms

This paper investigates the orbital stability of solitary waves for the generalized symmetric regularized-long-wave equations with two nonlinear terms and analyzes the influence of the interaction between two nonlinear terms on the orbital stability. SinceJis not onto, Grillakis-Shatah-Strauss theory cannot be applied on the system directly. We overcome this difficulty and obtain the general conclusion on orbital stability of solitary waves in this paper. Then, according to two exact solitary waves of the equations, we deduce the explicit expression of discriminationd′′(c)and give several sufficient conditions which can be used to judge the orbital stability and instability for the two solitary waves. Furthermore, we analyze the influence of the interaction between two nonlinear terms of the equations on the wave speed interval which makes the solitary waves stable.

Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system

In this paper, consideration is given to the 3-coupled nonlinear Schrödinger system i∂∂tuj+∂2∂x2uj+∑i=13bij|ui|2uj=0, where uj are complex-valued functions of (x,t)∈R2, j=1,2,3, and bij are positive constants satisfying bij=bji. It will be shown first that if the symmetric matrix B=(bij) satisfies certain conditions, then ground-state solutions of the 3-coupled nonlinear Schrödinger system exist, and moreover, they are orbitally stable. The theory is then extended to include solitary waves as well. In particular, it will be shown that when a solitary wave is perturbed, the perturbed solution must stay close to a solitary-wave profile in which the translation and phase parameters are prescribed functions of time. Properties of these functions are then studied. This is a continuous work of our previous paper where the 2-coupled nonlinear Schrödinger system was considered.