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D.E. Pelinovsky and R.H.J. Grimshaw

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Nonlocal models for envelope waves in a stratified fluid

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Stud. Appl. Math. 97, 369-391 (1996)

**Abstract:**

A new, nonlocal evolution equation similar to the nonlinear
Schrodinger equation is derived for envelope waves in a
continuously stratified fluid by means of a
multiscale perturbation technique. This new equation
governs propagation of quasi-harmonic wave packets having
length scales much longer than the depth of the
density variations and much shorter than the total depth
of fluid. Generalizations of the nonlocal evolution equation
for a description of two-dimensional wave
modulations are also presented. The modulational stability
of small-amplitude waves is then investigated in the framework
of the derived equations. It is shown that
quasi-harmonic waves with the scales under consideration
are unstable with respect to oblique perturbations at certain
angles.

**Keywords:**

NONLINEAR SCHRODINGER EQUATION, AMPLITUDE INTERFACIAL WAVES,
INTERNAL WAVES, GRAVITY WAVES, MODULATIONAL INSTABILITY,
EVOLUTION EQUATION, STABILITY ANALYSIS, FINITE DEPTH,
WATER WAVES, SHEAR FLOWS