Stability Of Solitary Waves Research Articles

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.

Bifurcation to Instability Through the Lens of the Maslov Index

The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems (Cornwell and Jones in SIAM J Appl Dyn Syst 17(1):754–787, 2018). In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These standing waves are known to become unstable as parameters vary. Our goal is to see how this established stability criterion manifests itself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism for instability. We further suggest how this mechanism might be used to reveal new instabilities in singularly perturbed models.

Stability of Solitary Waves for the Modified Camassa-Holm Equation

Stability of Solitary Waves for the Modified Camassa-Holm Equation

Existence and orbital stability of standing waves for the 1D Schrödinger-Kirchhoff equation

In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.

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.

Book Review: Nonlinear Dirac equations: Spectral stability of solitary waves

Book Review: Nonlinear Dirac equations: Spectral stability of solitary waves

Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity

We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the \begin{document}$ {\bf{U}}(1) $\end{document} -invariant 1D Klein–Gordon equation coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.

Numerical investigation of solitary wave stability for quantum dissipative systems

We consider a simple model describing the interaction of a quantum particle with a vibrational environment, which eventually acts as a friction on the particle. This equation admits soliton-like solutions, and we numerically investigate their stability when subjected to a small initial impulsion. Our findings illustrate the analogies with the behavior of classical particles and the relevance of asymptotic models.

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 vs. scattering in the cubic-quintic Schrödinger equation

We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable.

No solitary waves in 2D gravity and capillary waves in deep water

A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Precisely, we consider incompressible, irrotational, infinite depth water wave equation, either with gravity and without surface tension, or without gravity but with surface tension. In both of these cases we prove that there are no solitary waves.

Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrodinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrodinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.

Existence and stability of solitary waves for the inhomogeneous NLS

In this paper, we identify necessary and sufficient conditions for the existence of appropriately localized waves for the inhomogeneous semi-linear Schrödinger equation driven by the subLaplacian dispersion operators (−Δ)s,0<s≤1. We construct these waves and we establish sharp asymptotics, both at the singularity 0 and for large values. We show the non-degeneracy of these waves. Finally, we provide spectral and orbital stability classification, under slightly more restrictive assumptions.

Solitary and periodic waves in collisionless plasmas: The Adlam-Allen model revisited.

We consider the Adlam-Allen (AA) system of partial differential equations, which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions-which reduce to the solitary wave in the limiting case of an infinite period-as well as rational solutions that are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg-de Vries equation. Such a reduction is not only another justification for the above solitary wave dynamics, but may also offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon via numerical experiments.

Stability of solitary waves of a nonlinear beam equation

We investigate the stability of both traveling wave and standing wave solutions of a fourth-order nonlinear beam equation. We establish conditions on the traveling wave speeds and standing wave frequencies that imply instability. We also use numerical methods to determine precise ranges of stability and instability in one and two dimensions.

Numerical analyses for spectral stability of solitary waves near bifurcation points

We present a numerical approach for determination of the spectral stability of solitary waves by computing eigenvalues and eigenfunctions of the corresponding eigenvalue problems, along with their continuation, for nonlinear wave equations in one space dimension. We illustrate the approach for the nonlinear Schrodinger (NLS) equation with a small potential, and numerically determine the spectral stability of solitary waves near bifurcation points along with computations of eigenfunctions and eigenvalues. The numerical results demonstrate some theoretical ones the authors recently obtained for the example.

Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube

We first characterize strain solitary waves propagating in a fluid-filled membrane tube when the fluid is stationary prior to wave propagation and the tube is also subjected to a finite stretch. We consider the parameter regime where all traveling waves admitted by the linearized governing equations have nonzero speed. Solitary waves are viewed as waves of finite amplitude that bifurcate from the quiescent state of the system with the wave speed playing the role of the bifurcation parameter. Evolution of the bifurcation diagram with respect to the pre-stretch is clarified. We then study the stability of solitary waves for a representative case that is likely of most interest in applications, the case in which solitary waves exist with speed c lying in the interval [0, c_1) where c_1 is the bifurcation value of c, and the wave amplitude is a decreasing function of speed. It is shown that there exists an intermediate value c_0 in the above interval such that solitary waves are spectrally stable if their speed is greater than c_0 and unstable otherwise.

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.

Solitary waves in boundary layer induced by a travelling wave with increasing amplitude

The stability of solitary waves in an incompressible boundary layer, induced by a driving wave on the wall, is studied in order to investigate the generation and development of solitary waves. The nonlinear waves in the boundary layers, excited by a travelling wave with an increasing amplitude, rather than a standing wave, are described by forced Benjamin–Ono equation. In the study, pseudo-spectral method is used to approach to the solution of the governing equation. The results show that there are several wave energy wave branches and energy jumps to higher branch via violent oscillation. The wave pattern remains similar at every branch but the number of spike increases by one with every wave oscillation. In order to study the stability of the nonlinear waves, the perturbed wave equation is derived and a complex oscillator system is obtained from the linear perturbed wave equation. The stability analysis shows that the original wave loses its stability and wave energy begins to oscillate, and a new spike-like wave is born, when imaginary part of eigenvalue reaches to zero. Indeed, spile-like solitary waves can be induced by a travelling wave during the wave energy oscillations in boundary layers.