Open AccessExact elliptic compactons in generalized Korteweg–De Vries equations
Author(s) -
Cooper Fred,
Khare Avinash,
Saxena Avadh
Publication year - 2006
Publication title -
complexity
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 61
eISSN - 1099-0526
pISSN - 1076-2787
DOI - 10.1002/cplx.20133
Subject(s) - momentum (technical analysis) , mathematical physics , korteweg–de vries equation , nonlinear system , amplitude , physics , action (physics) , mathematics , mathematical analysis , stability (learning theory) , stability theorem , energy (signal processing) , quantum mechanics , cauchy distribution , finance , machine learning , computer science , economics
Using the action principle, and assuming a solitary wave of the generic form u ( x , t ) = AZ (β( x + q ( t )), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg‐De Vries equation K *( l , p ). Specifically we find that ${\dot q}=r(l,p)H/P$ , where l , p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two‐parameter family of exact solutions to these equations, which include elliptic and hyper‐elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria. © 2006 Wiley Periodicals, Inc. Complexity 11: 30–34, 2006