Stability of solitary waves of some coupled systems

Abstract

Solitary waves of Hamiltonian dispersive systems arise as critical points of the augmented Lagrangian V(u) + γI1(u) + ωI2(u), where V(u), I1(u) and I2(u) are first integrals of the evolution system (the case of two first integrals is also considered in this paper). According to variational methods, to show the stability of such solitary waves we have to show that the critical point is actually a local minimizer of the corresponding constrained variational problem. The major difficulty in applying this abstract method is to verify certain qualitative properties of the spectrum of the selfadjoint linearized operator M, which is the second derivative of the augmented Lagrangian at the critical point. In the examples we consider in this paper, the linearized operator M will be a linear selfadjoint ordinary differential operator given by a 2 × 2 system. Our main result states that, under some conditions, M has zero as a simple eigenvalue and it has exactly one negative eigenvalue. These are precisely the qualitative properties needed for the stability analysis using variational methods. We apply this result to study the stability of solitary waves for the following systems: a Schrödinger system from nonlinear optics (the so-called χ2-SHG equations), the (nonintegrable) Hirota–Satsuma system and a coupled system of a Schrödinger equation and a KdV equation.