Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations

Abstract

Considered here are systems of partial di erential equations arising in internal wave theory. The systems are asymptotic models describing the two- way propagation of long-crested interfacial waves in the Benjamin-Ono and the Intermediate Long-Wave regimes. Of particular interest will be solitary-wave solutions of these systems. Several methods of numerically approximating these solitary waves are put forward and their performance compared. The output of these schemes is then used to better understand some of the fundamental properties of these solitary waves. The spatial structure of the systems of equations is non-local, like that of their one-dimensional, unidirectional relatives, the Benjamin-Ono and the Intermediate Long-Wave equations. As the non-local aspect is comprised of Fourier multiplier operators, this suggests the use of spectral methods for the discretization in space. Three iterative methods are proposed and implemented for approximating traveling-wave solutions. In addition to Newton-type and Petviashvili iterations, an interesting wrinkle on the usual Petviashvili method is put forward which appears to o er advantages over the other two techniques. The performance of these methods is checked in several ways, including using the approximations they generate as initial data in time-dependent codes for obtaining solutions of the Cauchy problem. Attention is then turned to determining speed versus amplitude relations of these families of waves and their dependence upon parameters in the models. There are also provided comparisons between the unidirectional and bidirec- tional solitary waves. It deserves remark that while small-amplitude solitary- wave solutions of these systems are known to exist, our results suggest the amplitude restriction in the theory is arti cial.