The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: I. Initial data has a discontinuous expansive step

Nonlinearity ◽  
2008 ◽  
Vol 21 (10) ◽  
pp. 2391-2408 ◽  
Author(s):  
J A Leach ◽  
D J Needham
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Large Time ◽  
Vries Equation ◽  
Time Development ◽  
Initial Value ◽  
De Vries

The initial-value problem for the Korteweg-de Vries equation

1975 ◽  
Vol 278 (1287) ◽  
pp. 555-601 ◽  
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Model Equation ◽  
Vries Equation ◽  
Initial Value ◽  
Exact Relation ◽  
De Vries ◽  
Periodic Initial Data ◽  
Wave Limit ◽  
Continuous Dependence Results

For the Korteweg-de Vries equation u t + u x + u u x + u x x x = 0 , existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞< x <∞) and the periodic initial-value problem (posed on 0 ⩽ x ⩽ l with periodic initial data). The results are sharper than those obtained previously in that the solutions provided have the same number of L 2 -derivatives as the initial data and these derivatives depend continuously on time, as elements of L 2 . The same equation with dissipative and forcing terms added is also examined. A by-product of the methods used is an exact relation between solutions of the Korteweg-de Vries equation and solutions of an alternative model equation recently studied by Benjamin, Bona & Mahony (1972). It is proven that in the long-wave limit under which these equations are generally derived, the solutions of the two models posed for the same initial data are the same. In the process of carrying out this programme, new results are obtained for the latter model equation.

scholarly journals On the Korteweg–de Vries equation: Frequencies and initial value problem

1997 ◽  
Vol 181 (1) ◽  
pp. 1-55 ◽  
Author(s):  
D. Bättig ◽  
T. Kappeler ◽  
B. Mityagin
Keyword(s):  
Initial Value Problem ◽  
Vries Equation ◽  
Initial Value ◽  
De Vries
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