Nonlinear Wave Equations for Oceanic Internal Solitary Waves

2015 ◽  
Vol 136 (2) ◽  
pp. 214-237 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Nonlinear Wave ◽  
Solitary Waves ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Internal Solitary Waves

scholarly journals APPROXIMATE TECHNIQUES FOR DISPERSIVE SHOCK WAVES IN NONLINEAR MEDIA

2012 ◽  
Vol 21 (03) ◽  
pp. 1250035 ◽  
Author(s):  
T. R. MARCHANT ◽  
NOEL F. SMYTH
Keyword(s):  
Shock Wave ◽  
Approximate Method ◽  
Nonlinear Wave ◽  
Solitary Waves ◽  
Wave Equations ◽  
Leading Edge ◽  
Nonlinear Wave Equations ◽  
Trailing Edge ◽  
Nonlinear Media ◽  
Linear Waves

Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained.

scholarly journals A uniqueness result for the Sine-Gordon breather

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Rainer Mandel
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Uniqueness Result ◽  
Nonlinear Wave Equations

AbstractIn this note we prove that the sine-Gordon breather is the only quasimonochromatic breather in the context of nonlinear wave equations in $$\mathbb {R}^N$$ R N .

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