Near Homoclinic Orbits of the Focusing Nonlinear Schroedinger Equation
The homoclinic orbits of the integrable nonlinear Schrödinger equation are of interest because of the chaotic behaviour of the equation when it is damped and driven near such an orbit (McLaughlin D W and Overman E A 1995 Whiskered tori for integrable Pde's: chaotic behaviour in near integrable Pde's Surveys in Applied Mathematics vol 1 ed Keller et al (New York: Plenum) ch 2). It is known that exact expressions can be obtained for these homoclinic orbits via Bäcklund transformations using squared eigenfunctions of the associated linear operator (see above reference). In this paper, the stationary equations of the NLS hierarchy are used to separate the temporal and spatial flows into completely integrable finite-dimensional systems which define the saturation of a given linear instability of a plane wave. Near the homoclinic orbits satisfying even boundary conditions, the phase space is a product of two-dimensional phase planes similar to the phase plane of the Duffing oscillator. The near homoclinic orbits of the plane wave are realized numerically by a simple Runge-Kutta scheme that solves the two systems along the characteristic directions of space and time.
Wright, O. C. (1999). Near Homoclinic Orbits of the Focusing Nonlinear Schroedinger Equation. Nonlinearity, 12 (5), 1277-1287.