Asymptotics beyond All Orders pp 299-307 | Cite as

# Generalized Solitary Waves in a Stratified Fluid

## Abstract

The problems considered in this paper deal with generalized solitary waves in a stratified fluid over a flat bottom in the presence of surface tension. By a generalized solitary wave we mean a solitary wave with a small amplitude oscillation at infinity. First let us briefly discuss some recent progress in the exact theory of solitary waves on water of constant density with surface tension. An approximate equation governing solitary waves with surface tension was originally derived by Korteweg and de Vries^{1} and has been named after them. Let *τ* be the Bond number, a nondimensional surface tension coefficient, and *F* be the Froude number, the square of a nondimensional wave speed. The critical values of *τ* and *F* are respectively 1/3 and 1. For *τ* > 1/3, *F* < 1 but near 1, a solitary wave solution to this equation represents a wave of depression, and for 0< *τ* < 1/3, *F >* 1 but near 1, the solution denotes a wave of elevation. The existence of a solitary wave of depression, which decays to zero at infinity, was proved by Amick and Kirchgässner^{2}. For 0 < *τ* < 1/3, the linearized governing equations possess one positive eigenvalue and the corresponding eigenfunction is oscillatory. Numerical studies and other physical models due to Hunter and Vanden-Broeck^{3}, Zuferia^{4} and Boyd^{5} indicate that a generalized solitary wave may take place. The main difficulties are to isolate the oscillatory part and to estimate its amplitude. The existence proof of a generalized solitary wave has been given by Beale^{6} and Sun^{7} independently using different methods.

## Keywords

Surface Tension Solitary Wave Solvability Condition Positive Eigenvalue Solitary Wave Solution## Preview

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## References

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