A generalized Korteweg-de Vries equation in a quarter plane

1999 ◽  
pp. 59-125 ◽  
Author(s):  
Jerry L. Bona ◽  
Laihan Luo
Keyword(s):  
Vries Equation ◽  
Quarter Plane ◽  
De Vries

Asymptotics of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane

2020 ◽  
pp. 1-46
Author(s):  
Nan Liu ◽  
Boling Guo
Keyword(s):  
Large Time ◽  
Vries Equation ◽  
Large Time Behavior ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Time Behavior ◽  
Transform Method ◽  
Quarter Plane ◽  
De Vries ◽  
Fifth Order

The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou.

Modulation theory solution for resonant flow over topography

1987 ◽  
Vol 409 (1836) ◽  
pp. 79-97 ◽  
Keyword(s):  
Plane Problem ◽  
Vries Equation ◽  
Simple Wave ◽  
Modulation Equations ◽  
Weakly Nonlinear ◽  
Numerical Studies ◽  
Quarter Plane ◽  
De Vries ◽  
Flow Over Topography ◽  
Wave Limit

The near-resonant flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit, this flow being governed by a forced Korteweg-de Vries equation. It is proved from the modulation equations for the Korteweg-de Vries equation, which apply away from the obstacle, that no steady state can form upstream of the obstacle. This has been noted from previous experimental and numerical studies. The solution upstream and downstream of the topography is constructed as a simple wave solution of the modulation equations. Based on similarities between the method by which this solution is found and the quarter plane problem for the Korteweg-de Vries equation, the solution to the quarter plane problem is found for the special case in which a positive constant is specified at x = 0.

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