Purely nonlinear instability of standing waves with minimal energy

Abstract

We consider Hamiltonian systems with U (1) symmetry. We prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable, in spite of the absence of linear instability. Essentially, the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system. We apply our theory to the nonlinear Schrödinger equation.

Nonlinear Instability of a Critical Traveling Wave in the Generalized Korteweg–de Vries Equation

“…We address this question in this paper, proving the instability under certain rather generic assumptions. This result is the analog of [CP03] for the generalized Korteweg -de Vries equation (1.1).…”

Spectra of Linearized Operators for NLS Solitary Waves

“…They use an abstract projection (Riesz projection) onto the discrete spectrum to reduce the problem to a 4x4 matrix problem (and exploit the complex structure), while we are more direct. We thank the referee for pointing out [6] to us. Theorem 2.6 There are small constants µ * > 0 and ε * > 0 so that for every…”

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

“…In addition to the aforementioned references, there is of course a vast literature on the (in)stability of standing waves for nonlinear evolution equations. The orbital stability question (for Klein-Gordon, NLS, as well as many other classes of PDE) was addressed by Shatah [46], Shatah, Strauss [47], Weinstein [55], [56], Grillakis, Shatah, Strauss [22], [23] (who developed an "abstract" theory of orbital stability), Grillakis [21], Comech, Pelinovsky [9]. As far as the question of asymptotic stability is concerned (which is much closer to the present paper), see Soffer, Weinstein [49], [50], Buslaev, Perelman [5], [6], Cuccagna [10], Rodnianski, Schlag, Soffer [42], [43], Perelman [35], [36], [37], Fröhlich, Jonsson, Gustafson, Sigal [16], Fröhlich, Tsai, Yau [17], as well as Gang, Sigal [18], [19].…”