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

The higher-order nonlinear Schrödinger equation (NLS) (Dysthe’s equation in the context of water waves) models the time evolution of the slowly modulated amplitude of a wave packet in physical systems described by dispersive partial differential equations (PDEs). 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 this parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This article describes a procedure to derive a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. This is achieved by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Fréchet 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 and does not assume any additional structure to the governing PDE, for example its Hamiltonian nature. To complement this, two specific examples with accompanying symbolic algebra code are demonstrated that can be used as a template for other physical systems.