N-soliton formulae for theintermediate nonlinear Schrödinger equation

Abstract

A direct proof is given to the N-soliton solution of theintermediate nonlinear Schrödinger (INLS) equation describingenvelope waves. The proof relies only on an elementary theory ofdeterminants and knowledge of the inverse scatteringtransform method is not required. In particular, when theN-soliton solution is substituted into the system of bilinearequations for the INLS equation, the system is found to reduceto Jacobi's formula for determinants. A special class ofN-soliton solutions is also presented which is expressed interms of exponential functions. In the deep-water limit, thesolution reduces to the algebraic N-soliton solution of anonlocal NLS equation with the Hilbert kernel whereas in theshallow-water limit, the solution reduces to the N-solitonsolution of the defocusing NLS equation.