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Damping of Periodic Waves in Physically Significant Wave Systems
Author(s) -
Ablowitz M. J.,
Ablowitz S. A.,
Antar N.
Publication year - 2008
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2008.00419.x
Subject(s) - elliptic function , korteweg–de vries equation , mathematical analysis , nonlinear system , mathematics , amplitude , soliton , function (biology) , jacobi elliptic functions , physics , classical mechanics , quantum mechanics , evolutionary biology , biology
Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m = m ( t ) . The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m = 0 to soliton waves when m = 1 .