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Register a new userInstability of the solitary wave solutions for the genenalized derivative Nonlinear Schrodinger equation in the critical frequency case
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We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrodinger equation $$ ipartial_{t}u+partial_{x}^{2}u+i|u|^{2sigma}partial_x u=0. $$ The equation has a two-parameter family of solitary wave solutions of the form begin{align*} phi_{omega,c}(x)=varphi_{omega,c}(x)exp{big{ ifrac c2 x-frac{i}{2sigma+2}int_{-infty}^{x}varphi^{2sigma}_{omega,c}(y)dybig}}. end{align*} Here $ varphi_{omega,c}$ is some real-valued function. It was proved in cite{LiSiSu1} that the solitary wave solutions are stable if $-2sqrt{omega }<c <2z_0sqrt{omega }$, and unstable if $2z_0sqrt{omega }<c <2sqrt{omega }$ for some $z_0in(0,1)$. We prove the instability at the borderline case $c =2z_0sqrt{omega }$ for $1<sigma<2$, improving the previous results in cite{Fu-16-DNLS} where $3/2<sigma<2$.
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In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesnt enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4omega>c^2$ and the critical parameters $4omega=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters $4omega=c^2, cleq 0$ and the supercritical parameters $4omega<c^2$, there is no nontrivial solitary wave for (DNLS). At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for (DNLS) with initial data in the invariant set $mathcal{K}^+_{omega,c}subseteq H^1(R)$, with $4omega=c^2, c>0$ or $4omega>c^2$. On one hand, different with the scattering result for the $L^2$-critical NLS in cite{Dod:NLS_sct}, the scattering result of (DNLS) doesnt hold for initial data in $mathcal{K}^+_{omega,c}$ because of the existence of infinity many small solitary/traveling waves in $mathcal{K}^+_{omega,c},$ with $4omega=c^2, c>0$ or $4omega>c^2$. On the other hand, our global result improves the global result in cite{Wu-DNLS, Wu-DNLS2} (see Corollary ref{cor:gwp}).
An explicit lifespan estimate is presented for the derivative Schrodinger equations with periodic boundary condition.
We consider the derivative nonlinear Schrodinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is whether ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction $M(q)=int |q|^2 < 4pi$. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well-posed for initial data in $H^{1/6}$ under the same restriction on $M$. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.
We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrodinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.
We consider the nonlinear Klein-Gordon equation in $R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
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