Instability Of Solitary Wave Solutions Research Articles

Instability of solitary wave solutions for the nonlinear Schrödinger equation of derivative type in degenerate case

We study the stability theory of solitary wave solutions for a type of derivative nonlinear Schrödinger equation i∂tu+∂x2u+i|u|2∂xu+b|u|4u=0,b>0.The equation has a two-parameter family of solitary wave solutions of the form uω,c(x,t)=exp{iωt+ic2(x−ct)−i4∫−∞x−ct|φω,c(η)|2dη}φω,c(x−ct).Here φω,c is some suitable function and −2ω<c≤2ω. The stability in the frequency region of −2ω<c<2κω (for some κ∈(0,1)), and the instability in the frequency region of 2κω<c<2ω were proved by Ohta (2014). Recently, in the endpoint case c=2ω, the instability of uω,c was proved by Ning et al. (2017). Then the stability and instability region has been established except the degenerate case c=2κω. In this paper, we address the problem and prove its instability in the degenerate case.

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.

Instability of solitary wave solutions for derivative nonlinear Schrödinger equation in endpoint case

We study the stability theory of solitary wave solutions for a type of the derivative nonlinear Schrödinger equationi∂tu+∂x2u+i|u|2∂xu+b|u|4u=0. The equation has a two-parameter family of solitary wave solutions of the formeiω0t+iω12(x−ω1t)−i4∫−∞x−ω1t|φω(η)|2dηφω(x−ω1t). The stability theory in the frequency region of |ω1|<2ω0 was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case ω1=2ω0, in which the elliptic equation of φω is “zero mass”.

Instability of solitary waves for nonlinear Schrödinger equations of derivative type

We study the orbital stablity and instability of solitary wave solutions for nonlinear Schr\odinger equations of derivative type.

Instability of solitary waves of the generalized higher-order KP equation

This paper deals with the generalized higher-order Kadomtsev–Petviashvili (KP) equation. The strong instability of solitary wave solutions of this equation will be proved.

Instability of solitary wave solutions for the generalized BO–ZK equation

In this paper we study the generalized BO–ZK equation in two space dimensions u t + u p u x + α H u x x + ε u x y y = 0 . We review the existence theory for solitary waves and prove that they are nonlinearly unstable if p belongs to the range 4 / 3 < p < 4 . We also establish Strichartz-type estimates.

Blow up and instability of solitary wave solutions to a generalized Kadomtsev–Petviashvili equation and two-dimensional Benjamin–Ono equations

Let with p being the ratio of an even to an odd integer. For the generalized Kadomtsev–Petviashvili equation, coupled with Benjamin–Ono equations, in the form it is proved that the solutions blow up in finite time even for those initial data with positive energy. As a by-product, it is proved that for all , the solitary waves are strongly unstable if . This result, even in a special case , improves a previous work by Liu (Liu 2001 Trans. AMS 353 , 191–208) where the instability of solitary waves was proved only in the case of .

Instability of solitary waves for a generalized Benney–Luke equation

We study linear instability of solitary wave solutions of a one-dimensional generalized Benney–Luke equation, which is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension. Further, we implement a finite difference numerical scheme which combines an explicit predictor and an implicit corrector step to compute solutions of the model equation which is used to validate the theory presented.

Instability of solitary wave solutions to long-wavelength transverse perturbationsin the generalized Kadomtsev-Petviashvili equation with negative dispersion.

The present paper is a more detailed article of the previously published Letter [Phys. Rev. Lett. 84, 3065 (2000)]], which discovered the transversely unstable solitary wave solutions of the generalized Kadomtsev-Petviashvili (GKP) equation with negative dispersion. In addition to detailed explanation of stability analysis to long-wavelength transverse perturbations, numerical calculation of the GKP equation is carried out here to study the transverse stability to perturbations of finite wavelength. The numerical results show that there is a short-wavelength cutoff to the transverse instability. Moreover, we reveal the existence of transversely unstable solitons.

Linear Instability of Solitary Wave Solutions of the Kawahara Equation and Its Generalizations

The linear stability problem for solitary wave states of the Kawahara---or fifth-order KdV-type---equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multisymplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including changes in the state at infinity represented by embedding the solitary wave in a multiparameter family. The theory is used to give a rigorous geometric sufficient condition for instability. The theory is abstract and applies to a wide range of solitary wave states. For example, the theory is applied to the families of solitary waves found by Kichenassamy--Olver and Levandosky.

1113 一般化KP方程式による孤立波解の不安定性(O.S.11-3 乱流現象とEFD(その3))(O.S.11 乱流現象とEFD)

1113 一般化KP方程式による孤立波解の不安定性(O.S.11-3 乱流現象とEFD(その3))(O.S.11 乱流現象とEFD)

Stability and instability of solitary waves for generalized symmetric regularized-long-wave equations

This paper studies the orbital stability and instability of solitary wave solutions of the generalized symmetric regularized-long-wave equations. It is show that a traveling wave may be stable or unstable depending on the nonlinearity and the range of the wave's speed of propagation. Sharp conditions of that effect are given.