Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-field. Using various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (fifth-order) KdV equation, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-field.

[1]  Roger Grimshaw,et al.  Weakly Nonlocal Solitary Waves in a Singularly Perturbed Korteweg-De Vries Equation , 1995, SIAM J. Appl. Math..

[2]  Wei Hua,et al.  A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques , 1999, Appl. Math. Comput..

[3]  Prabir Daripa,et al.  A class of model equations for bi-directional propagation of capillary–gravity waves , 2003 .

[4]  J. Vanden-Broeck.  Elevation solitary waves with surface tension , 1991 .

[5]  P. Zweifel.  Advanced Mathematical Methods for Scientists and Engineers , 1980 .

[6]  Alfred Ramani,et al.  Structural stability of the Korteweg-De Vries solitons under a singular perturbation , 1988 .

[7]  J. Toland,et al.  Solitary waves with surface tension I: Trajectories homoclinic to periodic orbits in four dimensions , 1992 .

[8]  Harvey Segur,et al.  Asymptotics beyond all orders in a model of crystal growth , 1991 .

[9]  R. Grimshaw,et al.  The generation of radiating waves in a singularly-perturbed Korteweg-de Vries equation , 1993 .

[10]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[11]  John K. Hunter,et al.  Existence of perturbed solitary wave solutions to a model equation for water waves , 1988 .

[12]  S. Sun.  Existence of a generalized solitary wave solution for water with positive bond number less than 13 , 1991 .

[13]  Wiktor Eckhaus.  Singular perturbations of homoclinic orbits in R 4 , 1992 .

[14]  D. Peregrine.  A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95. , 1998, Journal of Fluid Mechanics.

[15]  J. Thomas Beale,et al.  Exact solitary water waves with capillary ripples at infinity , 1991 .

[16]  W. Eckhaus.  On water waves at Froude number slightly higher than one and Bond number less than 1/3 , 1992 .

[17]  Roger H.J. Grimshaw,et al.  Solitary internal waves with oscillatory tails , 1992, Journal of Fluid Mechanics.

[18]  J. Vanden-Broeck,et al.  Long periodic internal waves , 1992 .

[19]  J. Byatt-Smith.  On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter , 1991, European Journal of Applied Mathematics.

[20]  T. R. Akylas,et al.  On Short-Scale Oscillatory Tails of Long-Wave Disturbances , 1995 .

[21]  John P. Boyd,et al.  Weakly non-local solutions for capillary-gravity waves: fifth-degree Korteweg-de Vries equation , 1991 .