Instability Of Solitary Waves Research Articles

Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

Instability of solitary waves of two-dimensional Benjamin equation

Instability of solitary waves of two-dimensional Benjamin equation

Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

<p style='text-indent:20px;'>We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.</p>

Instability and blow-up of solutions of the fifth-order KP equation

In this paper, we study the dynamical behavior of solutions of the fifth-order Kadomtsev–Petviashvili equation. We improve the results of the previous works and show strong instability of solitary waves when the coefficient of the third-order dispersion term is positive. In spite of the lack of scaling, by using the variational characteristics of the solitary waves we obtain sharp thresholds for blow-up and global existence by means of new estimates.

Strong instability of solitary waves for inhomogeneous nonlinear Schrödinger equations

This paper studies the inhomogeneous nonlinear Schrödinger equations, which may model the propagation of laser beams in nonlinear optics. Using the cross‐constrained variational method, a sharp condition for global existence is derived. Then, by solving a variational problem, the strong instability of solitary waves of this equation is proved.

Multi‐dimensional instability of electrostatic solitary waves in a warm multi‐ion dust plasma

AbstractStudy of dust ion acoustic waves in a magnetized dusty plasmas composed of negatively or positively charged static dust, positive and negative ions, as well as kappa distribution electrons is presented. The Zakharov–Kuznetsov (ZK) equation is derived via reductive perturbation technique. The solitary wave solution of ZK equation is given and the multi‐dimensional instability of these solitary waves is investigated via small k perturbation method. The instability criterion and growth rate relying on obliqueness, superthermality, positive ion thermal pressure, relative ion number density, magnetic field strength, and direction cosines are discussed for five cases. The results are beneficial to understand different nonlinear characteristics of unstable electrostatic disturbances in laboratory and space plasmas.

Well-Posedness for a Coupled System of Kawahara/KdV Type Equations

We consider the initial value problem (IVP) associated to a coupled system of Kawahara/KdV type equations. We prove the well-posedness results for given data in a Gevrey spaces. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important. The techniques presented in this work were based in Grujic and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation, whose the initial data belongs to Gevrey spaces.

On the Stability of Solitary Water Waves with a Point Vortex

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small‐amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite‐dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS) , but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state‐dependent.As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized Korteweg–de Vries (KdV) and Benjamin‐Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss , who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion and obtain a new instability result for fractional KdV. © 2020 Wiley Periodicals, Inc.

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 and Instability of Solitary Waves for a Class of Dispersive Symmetric Regularized Long-Wave Equation

This paper studies the orbital stability and instability of solitary wave solutions of the generalized symmetric regularized long-wave-type equations. It is shown that a traveling wave may be stable or unstable depending on the range of the waves speed of propagation and the relation between the dispersion (i.e., the order of the pseudodifferential operator) and the nonlinearity.

Slug generation processes in co-current turbulent-gas/laminar-liquid flows in horizontal channels

We theoretically and computationally investigate the physical processes of slug-flow development in concurrent two-phase turbulent-gas/laminar-liquid flows in horizontal channels. The objective is to understand the fundamental mechanisms governing the initial growth and subsequent nonlinear evolution of interfacial waves, starting from a smooth stratified flow of two fluids with disparity in density and viscosity and ultimately leading to the formation of intermittent slug flow. We numerically simulate the entire slug development by means of a fully coupled immersed flow (FCIF) solver that couples the two disparate flow dynamics through an immersed boundary (IB) method. From the analysis of spatial/temporal interface evolution, we find that slugs develop through three major cascading processes: (I) stratified-to-wavy transition; (II) development and coalescence of long solitary waves; and (III) rapid channel bridging leading to slugging. In Process I, relatively short interfacial waves form on the smooth interface, whose growth is governed by the Orr–Sommerfeld instability. In Process II, interfacial waves evolve into long solitary waves through multiple resonant and near-resonant wave–wave interactions. From instability analysis of periodic solitary waves, we show that these waves are unstable to their subharmonic disturbances and grow in amplitude and primary wavelength through wave coalescence. The interfacial forcing from the turbulent gas–laminar liquid interactions significantly precipitates the growth of instability of solitary waves and enhances coalescence of solitary waves. In Process III, we show by an asymptotic analysis that interfacial waves achieve multiple-exponential growth right before bridging the channel, consistent with observations in existing experiments. The present study provides important insights for effective modelling of slug-flow dynamics and the prediction of slug frequency and length, important for design and operation of (heavy-oil/gas) pipelines and production facilities.

Stability properties of solitary waves for fractional KdV and BBM equations

This paper sheds new light on the stability properties of solitary wave solutions associated with Korteweg–de Vries-type models when the dispersion is very low. Using a compact, analytic approach and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of spectral instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and spectral instability of the ground state solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis, we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation.The arguments presented in this investigation show promise for use in the study of the instability of traveling wave solutions of other nonlinear evolution equations.

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.

Instability of solitary waves for a generalized derivative nonlinear Schrödinger equation in a borderline case

We study the orbital instability of solitary waves for a derivative nonlinear Schrodinger equation with a general nonlinearity. We treat a borderline case between stability and instability, which is left as an open problem by Liu, Simpson and Sulem (2013). We give a sufficient condition for instability of a two-parameter family of solitary waves in a degenerate case by extending the results of Ohta (2011), and verify this condition for some cases.

Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity

The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed.Grillakis, Shatah and Strauss’ stability theory is applied for the investigation of the orbital stability or instability of solitary waves to the double dispersion equation. An analytical formula, related to some conservation laws of the problem, is derived. As a consequence, the dependence of orbital stability or instability on the parameters of the problem is demonstrated. A complete characterization of the values of the wave velocity, for which the solitary waves to the generalized Boussinesq equation are orbitally stable or unstable, is given.In the special case of a quadratic nonlinearity our results are reduced to those known in the literature.

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.

Stability and Instability of Improved Heimburg{Jackson Model to Nerve Pulse Propagation

There are a number of mathematical models to nerve pulse propagation in biomembranes, as Hodgkin--Huxley, FitzHugh--Nagumo and Heimburg--Jackson models, see, e.g., [1,2]. However, these models do not describe adequately all observed phenomena.Recently in [2], generalized Boussinesq equation with quadratic--cubic nonlinearity is proposed as an improvement of the well-known Heimburg--Jackson model.In this study we prove analytically the orbital stability and instability of solitary waves to the improved Heimburg--Jackson model (1). The results depend on the relationship between all parameters of the model. For the set of data, obtained experimentally, our theoretical results are in full agreement with the numerical simulations, presented in [3].

Transverse Instability of Dust-Acoustic Solitary Waves in Magnetized Dusty Plasmas**supported by National Natural Science Foundation of China (No. 11047010)

We theoretically investigated the transverse instability of three-dimensional (3D) dust-acoustic solitary waves in a magnetized dusty plasma. First, a 3D nonlinear Zakharov-Kuznetsov (ZK) equation, which can be used to describe the time-evolution of dust-acoustic solitary waves in magnetized dusty plasmas, is derived by using the reductive perturbation method. Second, we established a numerical scheme to study the transverse instability of the solitary waves described by the ZK equation. It was found that both stable and unstable solitary waves exist.

Vakhitov–Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

We study the linear stability of localized modes in self-interacting spinor fields, analysing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov–Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov–Kolokolov condition dQ/dω = 0 and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross–Neveu model and the generalized massive Thirring model in the charge-subcritical, charge-critical and charge-supercritical cases, illustrating the agreement with the Vakhitov–Kolokolov and the energy vanishing conditions.

Instability of ground states for a quasilinear Schrӧdinger equation

We study a class of one parameter (denoted by $\kappa$) family of quasi-linear Schrӧdinger equations arising in the theory of superfluid film in plasma physics. Using variational techniques, we prove the orbital instability of solitary waves for small values of the parameter $\kappa$, which gives an answer to a question raised in [9].