A Study of Well Posedness for Systems of Coupled Non-linear Dispersive Wave Equations

To model two-way propagation of waves in physical systems where nonlinear and dispersive effects are equally important, coupled systems of partial differential equations arise. The focus of this study is a particular coupled system of two evolution equations of generalized BBM-type. The coupling in these model systems is through the nonlinearities, which are a pair of homogenous quadratic polynomials. They have the same general mathematical structure as do more complicated models of surface and internal wave propagation. The present study is concerned with initial-value problems wherein the wave profile and the velocity are specified at a starting time. This is a natural generalization of the initial-value problem for the BBM or regularized long-wave (RLW) equation itself, that was originally proposed as an alternative to the classical Korteweg-de Vries (KdV) equation. Equations like KdV and BBM model unidirectional propagation of small amplitude, long wavelength waves. The initial condition represents a snapshot of an initial disturbance. The coupled systems allow for two-way propagation of waves, and so have a wider range of applicability. They are, however, mathematically more intricate. These coupled BBM-type systems are shown to be locally well-posed in the $L_2$-based Sobolev spaces $H^s$ for any $s \geq 0$. The further spatial and temporal regularity of solutions is also investigated. It transpires that there is no smoothing in the spacial variable, but there is smoothing in the temporal variable. Conditions are derived that imply the local well-posedness theory to extend globally. That is, under exact conditions on the coefficients of the quadratic nonlinearities, solutions are shown to exist and remain bounded for all time provided the initial data is at least in $H^1$. Using a Fourier splitting technique, global solutions are also inferred even when the initial data is only in $L_2$.