1999/04/30 by Bishwajyoti Dey, Avinash Khare · 10 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Dispersion (optics) #Dispersive partial differential equation #Geometry #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Order (exchange) #Physics #Quantum mechanics #Scaling #Stability (learning theory) #nlin.PS #nlin.SI #patt-sol #quant-ph #solv-int
paper · pdf · doi:10.1088/0305-4470/33/30/305
published in Journal of Physics A Mathematical and General 33(30), 5335-5344 (Institute of Physics) · 20 pages, To Appear in J.Phys.A (2000), several modifications
arxiv created 2000/07/02 · openalex publication_date 2000/07/19 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider fifth-order nonlinear dispersive K ( m , n , p ) type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves such as solitons, the interaction of the nonlinear dispersion with nonlinear convection generates compactons - the compact solitary waves free of exponential tails. This interaction also generates many other solitary wave structures such as cuspons, peakons, tipons etc which are otherwise unattainable with linear dispersion. Various self-similar solutions of these higher-order nonlinear dispersive equations are also obtained using similarity transformations. Further, it is shown that, like the third-order nonlinear K ( m , n ) equations, the fifth-order nonlinear dispersive equations also have the same four conserved quantities and, furthermore even any arbitrary odd-order nonlinear dispersive K ( m , n , p ,...) type equations also have the same three (and most likely the four) conserved quantities. Finally, the stability of the compacton solutions for the fifth-order nonlinear dispersive equations are studied using linear stability analysis. From the results of the linear stability analysis it follows that, unlike solitons, all the allowed compacton solutions are stable, since the stability conditions are satisfied for arbitrary values of the nonlinear parameters.