Stability Of Solitary Waves Research Articles

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.

Stability of Solitary Waves in Biomembranes and Nerves

The orbital stability or instability of solitary waves to the double dispersion equation, modeling the pules propagation in biomembranes and nerves, is investigated. An analytical formula, determining the regions of stability or instability of solitary waves, is derived. Solitary waves with high velocity are treated. The dependence of the regions of stability on the velocity and the parameters of the model is studied.

Existence and linearized stability of solitary waves for a quasilinear Benney system

We prove the existence of solitary wave solutions to the quasilinear Benney systemwhere , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

Asymptotic stability of solitary waves in generalized Gross–Neveu model

For the nonlinear Dirac equation in (1+1)D with scalar self-interaction (Gross–Neveu model), with quintic and higher order nonlinearities (and within certain range of the parameters), we prove that solitary wave solutions are asymptotically stable in the “even” subspace of perturbations (to ignore translations and eigenvalues ±2ωi). The asymptotic stability is proved for initial data in H1. The approach is based on the spectral information about the linearization at solitary waves which we justify by numerical simulations. For the proof, we develop the spectral theory for the linearized operators and obtain appropriate estimates in mixed Lebesgue spaces, with and without weights.

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.

Solitary shock waves and adiabatic phase transition in lipid interfaces and nerves.

This study shows that the stability of solitary waves excited in a lipid monolayer near a phase transition requires positive curvature of the adiabats, a known necessary condition in shock compression science. It is further shown that the condition results in a threshold for excitation, saturation of the wave's amplitude, and the splitting of the wave at the phase boundaries. Splitting in particular confirms that a hydrated lipid interface can undergo condensation on adiabatic heating, thus showing retrograde behavior. Finally, using the theoretical insights and state dependence of conduction velocity in nerves, the curvature of the adiabatic state diagram is shown to be closely tied to the thermodynamic blockage of nerve pulse propagation.

Dynamics of strongly nonlinear internal long waves in a three-layer fluid system

Dynamics of internal solitary waves in shallow water were studied with a strongly nonlinear internal long wave model in a three-layer fluid system. It is an extension of the two-layer MCC (Miyata, Choi, and Camassa) model, which is derived under the assumption that the characteristic wavelength is long in comparison to the thickness of each fluid layer. Phase velocity was assessed to understand the stability of solitary waves when they propagate. The instability mechanism turns out to be a Kelvin-Helmholtz type instability that is similar to the twolayer MCC model. We numerically investigated the three-layer model in terms of the effects of the middle layer and non-uniform bottom topography, and examined the generation of solitary waves to demonstrate the rich dynamics of the three-layer model.

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g(2)/κ+1(̅ΨΨ)(κ+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Ψ(x,t)=ψ(x)e(-iωt) for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t(c), it takes for the instability to set in, is an exponentially increasing function of ω and t(c) decreases monotonically with increasing κ.

Linear Stability of Solitary Waves for the One‐Dimensional Benney–Luke and Klein–Gordon Equations

The linear stability of the solitary waves for the one‐dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.

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.

Stability of solitary waves for a generalized higher-order Shallow water equation

In this work, we consider solitary wave solutions of a generalized higher-order shallow water equation. We investigate the existence and stability of solitary waves of the equation.

Scattering of solutions and stability of solitary waves for the generalized BBM-ZK equation

Scattering of solutions and stability of solitary waves for the generalized BBM-ZK equation

Stability of ion acoustic solitary waves in a magnetized plasma consisting of warm adiabatic ions and non-thermal electrons having vortex-like velocity distribution

AbstractSchamel's modified Korteweg-de Vries–Zakharov–Kuznetsov (S-ZK) equation, governing the behavior of long wavelength, weak nonlinear ion acoustic waves propagating obliquely to an external uniform static magnetic field in a plasma consisting of warm adiabatic ions and non-thermal electrons (due to the presence of fast energetic electrons) having vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space-independent) and static (time-independent) magnetic field, admits solitary wave solutions having a sech4 profile. The higher order stability of this solitary wave solution of the S-ZK equation has been analyzed with the help of multiple-scale perturbation expansion method of Allen and Rowlands (Allen, M. A. and Rowlands, G. 1993 J. Plasma Phys. 50, 413; 1995 J. Plasma Phys. 53, 63). The growth rate of instability is obtained correct to the order k2, where k is the wave number of a long wavelength plane wave perturbation. It is found that the lowest order (at the order k) instability condition is strongly sensitive to the angle of propagation (δ) of the solitary wave with the external uniform static magnetic field, whereas at the next order (at the order k2) the solitary wave solutions of the S-ZK equation are unstable irrespective of δ. It is also found that the growth rate of instability up to the order k2 for the electrons having Boltzmann distribution is higher than that of the non-thermal electrons having vortex-like distribution for any fixed δ.

Blow-up phenomena and stability of solitary waves for a generalized Dullin-Gottwald-Holm equation

In this work, we consider the Cauchy problem of the generalized Dullin-Gottwald-Holm equation. We establish a blow-up result for the generalized Dullin-Gottwald-Holm equation. In addition to this, we investigate the stability of solitary wave solutions of the equation.

Solitary matter waves in combined linear and nonlinear potentials: Detection, stability, and dynamics

We study statically homogeneous Bose-Einstein condensates with spatially inhomogeneous interactions and outline an experimental realization of compensating linear and nonlinear potentials that can yield constant-density solutions. We illustrate how the presence of a step in the nonlinearity coefficient can only be revealed dynamically and consider, in particular, how to reveal it by exploiting the inhomogeneity of the sound speed with a defect-dragging experiment. We conduct computational experiments and observe the spontaneous emergence of dark solitary waves. We use effective-potential theory to perform a detailed analytical investigation of the existence and stability of solitary waves in this setting, and we corroborate these results computationally using a Bogoliubov-de Gennes linear stability analysis. We find that dark solitary waves are unstable for all step widths, whereas bright solitary waves can become stable through a symmetry-breaking bifurcation as one varies the step width. Using phase-plane analysis, we illustrate the scenarios that permit this bifurcation and explore the dynamical outcomes of the interaction between the solitary wave and the step.

Stability of solitary waves for the vector nonlinear Schrödinger equation in higher-order Sobolev spaces

In this manuscript, a sharp form of orbital stability for the Schrödinger system, also known as the vector NLS, i∂∂tuj+∂2∂xxuj+2∑i=1m|ui|2uj=0, where uj are complex-valued functions of (x,t)∈R2, j=1,2,…,m, is established in L2-based Sobolev classes of arbitrarily high order. The result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave, but emerging residual oscillations must also be very small and not only in the energy norm.

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.

Existence and stability of solitary waves for the generalized Korteweg-de Vries equations

In this paper, we consider the fractional Korteweg-de Vries equations with general nonlinearities. By studying constrained minimization problems and applying the method of concentration-compactness, we obtain the existence of solitary waves for the generalized Korteweg-de Vries equations under some assumptions of the nonlinear term. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.

Local well-posedness and stability of solitary waves for the two-component Dullin–Gottwald–Holm system

In this paper, we study the Cauchy problem for the two-component Dullin–Gottwald–Holm system with the initial data (u0,ρ0)∈Bp,rs(R)×Bp,rs−1(R),1≤p,r≤+∞, and s>max{1+1p,32,2−1p}, which generalizes some previous local well-posed results in Sobolev spaces. Then we prove that all the smooth solitary wave solutions are orbitally stable under small disturbances.