On the connection of the solvability of a nonlinear evolution equation of the fourth order and elliptic integrals

2011 ◽  
Vol 11 (1-2) ◽  
pp. 45-56
Author(s):  
Alfred Huber
Keyword(s):  
Evolution Equation ◽  
Fourth Order ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Elliptic Integrals

scholarly journals Fourth Order Nonlinear Evolution Equation For Interfacial Gravity Waves In The Presence Of Air Flowing Over Water And A Basic Current Shear

2015 ◽  
Vol 20 (3) ◽  
pp. 517-530
Author(s):  
D.P. Majumder ◽  
A.K. Dhar
Keyword(s):  
Evolution Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Wave Number ◽  
Marginal Stability ◽  
Starting Point

Abstract A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

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