Temporal Evolutions and Stationary Waves for perturbed KdV equation with Nonlocal Term

Initial value problems as well as stationary solitary and periodic waves are investigated for a perturbed KdV equation including the Hilbert transform; ut + uux + βuxxx + η(ℋux - uxx) = 0 (η > 0). Multi-hump stationary solitary and periodic wave solutions are numerically identified. Furthermore, the close relation between the structure of the stationary waves and the behavior of the temporal evolutions is discussed in comparison with other perturbed KdV equations with different instability and dissipation terms. The results support some general features common to this type of nonlinear evolution equations.

[1]  Chaotic perturbations of Kdv. I. rational solutions , 1996 .

[2]  J. Eckmann,et al.  A global attracting set for the Kuramoto-Sivashinsky equation , 1993 .

[3]  Bao-Feng Feng,et al.  Temporal evolutions and stationary waves for dissipative Benjamin-Ono equation , 2000 .

[4]  Roger Temam,et al.  Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attr , 1985 .

[5]  Takuji Kawahara,et al.  Pulse interactions in an unstable dissipative‐dispersive nonlinear system , 1988 .

[6]  B. Feng,et al.  Stationary travelling-wave solutions of an unstable KdV-Burgers equation , 2000 .

[7]  N. Kudryashov,et al.  EXACT SOLITON SOLUTIONS OF THE GENERALIZED EVOLUTION EQUATION OF WAVE DYNAMICS , 1988 .

[8]  Takuji Kawahara,et al.  Formation of Saturated Solitons in a Nonlinear Dispersive System with Instability and Dissipation , 1983 .

[9]  I. Kevrekidis,et al.  Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations , 1990 .

[10]  Roger Temam,et al.  Inertial Manifolds for the Kuramoto-Sivashinsky Equation and an Estimate of their Lowest Dimension , 1986 .

[11]  B. Feng,et al.  Multi-hump stationary waves for a Korteweg-de Vries equation with nonlocal perturbations , 2000 .

[12]  A. Porubov Exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid , 1993 .

[13]  Silvina Ponce Dawson,et al.  Collections of heteroclinic cycles in the Kuramoto-Sivashinsky equation , 1997 .

[14]  Hsueh-Chia Chang,et al.  Wave evolution on a falling film , 1994 .

[15]  D. J. Benney Long Waves on Liquid Films , 1966 .

[16]  N. A. Kudryashov,et al.  Solitary waves in active-dissipative dispersive media , 1996 .

[17]  M. Taboada Finite-dimensional asymptotic behavior for the Swift-Hohenberg model of convection , 1990 .

[18]  L. Tsimring,et al.  Negative energy waves in hydrodynamics , 1986 .

[19]  Darryl D. Holm,et al.  Low-dimensional behaviour in the complex Ginzburg-Landau equation , 1988 .

[20]  Hsueh-Chia Chang,et al.  Laminarizing effects of dispersion in an active-dissipative nonlinear medium , 1993 .

[21]  H. McKean,et al.  Rational and elliptic solutions of the korteweg‐de vries equation and a related many‐body problem , 1977 .