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Register a new userInstability of nonlinear dispersive solitary waves
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Zhiwu Lin
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We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula are used to find the instability criteria. Recently, these techniques have also been extended to study instability of periodic waves and to the full water wave problem.
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We consider the linearized instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are not obtained at the highest wave, which has a 120 degree angle at the crest. Under the assumption of non-existence of secondary bifurcation which is confirmed numerically, we prove linear instability of solitary waves which are higher than the wave of maximal energy and lower than the wave of maximal travel speed. It is also shown that there exist unstable solitary waves approaching the highest wave. The unstable waves are of large amplitude and therefore this type of instability can not be captured by the approximate models derived under small amplitude assumptions. For the proof, we introduce a family of nonlocal dispersion operators to relate the linear instability problem with the elliptic nature of solitary waves. A continuity argument with a moving kernel formula is used to study these dispersion operators to yield the instability criterion.
We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schrodinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumption on the dispersive operator, we use a bootstrap argument to overcome the loss of derivative.
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}).
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.
We consider nonlinear Schrodinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
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