On varying coefficients of spatial inhomogeneous nonlinear Schr\"odinger equation

TL;DR

This paper revisits a spatial inhomogeneous nonlinear Schr"odinger equation modeling wave packets over varying topography, correcting previous coefficients and analyzing conditions where classical solitons cannot form due to alternating focusing and defocusing regimes.

Contribution

It provides analytical corrections to the nonlinear coefficient in the inhomogeneous NLS equation and explores the implications for soliton formation under varying topography.

Findings

Corrected the nonlinear coefficient for the inhomogeneous NLS equation.

Showed that the equation alternates between focusing and defocusing regimes.

Classical solitons do not form under certain topographical and wave conditions.

Abstract

A nonlinear evolution equation for wave packet surface gravity waves with variation in topography is revisited in this article. The equation is modeled by a spatial inhomogeneous nonlinear Schr\"odinger (NLS) equation with varying coefficients, derived by Djordjevi\'c and Redekopp (1978) and the nonlinear coefficient is later corrected by Dingemans and Otta (2001). We show analytically and qualitatively that the nonlinear coefficient and the corresponding averaging value, stated but not derived, by Benilov, Flanagan and Howlin (2005) and Benilov and Howlin (2006) are inaccurate. For a particular choice of topography and wave characteristics, the NLS equation alternates between focusing and defocusing case and hence, it does not admit the formation of a classical soliton, neither bright nor dark one.