Stability of Solitary Waves

* Introduction* The Miura transformation and resonant modes of the linearized operator* Semigroup estimates for the linearized KP-II equation* Preliminaries* Decomposition of the perturbed line soliton* Modulation equations* A priori estimates for the local speed and the local phase shift* The $L^2(\mathbb{R}^2)$ estimate* Decay estimates in the exponentially weighted space* Proof of Theorem 11* Proof of Theorem 14* Proof of Theorem 15* Appendix A Proof of Lemma 61* Appendix B Operator norms of $S^j_k$ and $\widetilde{C_k}$* Appendix C Proofs of Claims 62, 63 and 71* Appendix D Estimates of $R^k$* Appendix E Local well-posedness in exponentially weighted space* Bibliography

Stability of line solitons for the KP-II equation in R2

In this paper we consider the transverse instability for a nonlinear Schrodinger equation with power nonlinearity on R × TL, where 2πL is the period of the torus TL There exists a critical period 2πLω,p such that the line standing wave is stable for L Lω,p Here we farther study the bifurcation from the boundary L = Lω,p between the stability and the instability for line standing waves of the nonlinear Schrodinger equation We show the stability for the branch bifurcating from the line standing waves by applying the argument in Kirr, Kevrekidis and Pelinovsky [16] and the method in Grillakis, Shatah and Strauss [12] However, at the bifurcation point, the linearized operator around the bifurcation point is degenerate To prove the stability for the bifurcation point, we apply the argument in Maeda [18]

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Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations

In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient conditions for transverse instabilities to occur. This index is geometric in nature, and applies to any periodic traveling wave profile under some minor smoothness assumptions on the nonlinearity. We also describe the analogous theory for periodic traveling waves of the generalized Benjamin-Bona-Mahony equation to long wavelength transverse perturbations in the gBBM-Zakharov-Kuznetsov equation.

The Transverse Instability of Periodic Waves in Zakharov-Kuznetsov Type Equations

Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross--Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitons suffer from instability with respect to periodic transverse perturbations of large periods. The instability is induced by the spatial translation for the massive Thirring model and by the gauge rotation for the massive Gross--Neveu model. We also observe numerically that the instability holds for the transverse perturbations of any period in the massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the massive Gross--Neveu model.

Transverse instability of line solitons in massive Dirac equations

In this paper, we consider the stability for line solitary waves of the two dimensional Zakharov-Kuznetsov equation on $\mathbb{R}\times\mathbb{T}_L$ which is one of a high dimensional generalization of Korteweg-de Vries equation , where $\mathbb{T}_L$ is the torus with the period $2\pi L$. The orbital and asymptotic stability of the one soliton of Korteweg-de Vries equation on the energy space has been proved by Benjamin, Pego and Weinstein and Martel and Merle. We regard the one soliton of Korteweg-de Vries equation as a line solitary wave of Zakharov-Kuznetsov equation on $\mathbb{R}\times\mathbb{T}_L$. We prove the stability and the transverse instability of the line solitary waves of Zakharov-Kuznetsov equation by applying Evans' function method and the argument of Rousset and Tzvetkov. Moreover, we prove the asymptotic stability for the orbitally stable line solitary wave of Zakharov-Kuznetsov equation by using the argument of Martel and Merle, a Liouville type theorem and a corrected virial type estimate.

Stability for line solitary waves of Zakharov-Kuznetsov equation

Stability for line solitary waves of Zakharov–Kuznetsov equation

In this paper, we consider the transverse instability for a system of nonlinear Schrodinger equations on $\mathbb{R} \times \mathbb{T}_L $.
Here, $\mathbb{T}_L$ means the torus with a $2\pi L$ period.
It was shown by Colin-Ohta [11] that this system on $\mathbb{R}$ has a stable standing wave.
In this paper, we regard this standing wave as the standing wave of this system on $\mathbb{R} \times \mathbb{T}_L$.
Then, we show that there exists the critical period $L_{\omega}$ which is the boundary between the stability and the instability of the standing wave on $\mathbb{R} \times \mathbb{T}_L$.

Transverse instability for a system of nonlinear Schrödinger equations

In this paper, we consider the transverse instability for a nonlinear Schrodinger equation with a linear potential on ${\mathbb {R} \times \mathbb {T}_L}$, where $2\pi L$ is the period of the torus $\mathbb{T}_L$. Rose and Weinstein [18] showed the existence of a stable standing wave for a nonlinear Schrodinger equation with a linear potential. We regard the standing wave of nonlinear Schrodinger equation on ${\mathbb R}$ as a line standing wave of nonlinear Schrodinger equation on ${\mathbb R} \times {\mathbb T}_L$. We show the stability of line standing waves for all $L>0$ by using the argument of the previous paper [26].

Yohei Yamazaki

Papers

Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations

Stability for line solitary waves of Zakharov-Kuznetsov equation

Stability for line solitary waves of Zakharov–Kuznetsov equation

Transverse instability for a system of nonlinear Schrödinger equations

Transverse instability for nonlinear Schrödinger equation with a linear potential