Parameter-free higher-order Schrodinger systems with weak dissipation and forcing

Abstract

The higher-order nonlinear Schrodinger equation (Dysthe's equation in the context of water-waves) models the time evolution of the slowly modulated amplitude of a wave-packet in dispersive partial differential equations (PDE). These systems, of which water-waves are a canonical example, require the presence of a small-valued ordering parameter so that a multi-scale expansion can be performed. However, often the resulting system itself contains the small-ordering parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This paper derives a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. Instead of focusing on the water-wave problem or another specific problem, our procedure avoids the complicated algebra by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Frechet derivatives. An attractive feature of this procedure is that it can be used in many different physical settings, including water-waves, nonlinear optics and any dispersive system with weak dissipation or forcing. The paper concludes by discussing two specific examples.