Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves

Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves

Standing waves are a fundamental class of solutions of nonlinear wave equations with a spatial reflection symmetry, and they routinely arise in optical and oceanographic applications. At the linear level they are composed of two synchronized counterpropagating periodic traveling waves. At the nonlinear level, they can be defined abstractly by their symmetry properties. In this paper, general as...

Large scale instability of nonlinear standing waves

Multisymplectic geometry, covariant Hamiltonians, and water waves

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges. In this theory, solutions of a partial differential equation are sections of a fibre ...

Stability and instability of standing waves for nonlinear Schrödinger equations

Strong Instability of Standing Waves for Nonlinear Klein-gordon Equations

The strong instability of ground state standing wave solutions eφω(x) for nonlinear Klein-Gordon equations has been known only for the case ω = 0. In this paper we prove the strong instability for small frequency ω.