摘要:

A multiple‐scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple‐valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann‐Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple‐phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple‐valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearce

ISSN:0022-2526    

DOI:10.1002/sapm19857211

年代:1985

数据来源: WILEY

摘要:

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculatingN‐soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self‐truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering

ISSN:0022-2526    

DOI:10.1002/sapm198572139

年代:1985

数据来源: WILEY

摘要:

A recursion formula is described which generates infinite hierarchies of completely integrable Hamiltonian systems of nonlinear partial differential equations. These equations govern the evolution of a functionuofx,twhich takes its values in a semisimple Lie algebra. A Hamiltonian for the hierarchy is given in terms of a meromorphic connection matrix.

ISSN:0022-2526    

DOI:10.1002/sapm198572165

年代:1985

数据来源: WILEY

摘要:

Temporally growing modes of the linearized equations of motion for homogeneous shear flows in the beta‐plane are considered. A new upper bound on their rate of growth is derived. This bound is related to the necessary criterion for linear instability derived by Fjørtoft [1]. As a flow stabilizes due to increased beta‐effect or decreased basic‐state vorticity gradient, the upper bound on the growth rate decreases to zero. For more stable flows this newly derived bound is tighter than that derived by Høil

ISSN:0022-2526    

DOI:10.1002/sapm198572187

年代:1985

数据来源: WILEY