Exact Traveling-Wave Solutions to Bidirectional Wave Equations

AbstractIn this paper, we present several systematicways to find exact traveling-wave solutions of thesystems $$\eta _t + u_x + \left( {u\eta } \right)_x + au_{xxx} - b\eta _{xxt} = 0$$ $$u_t + \eta _x + uu_x + c\eta _{xxx} + du_{xxt} = 0$$ where a, b, c, and d are real constants. These systems,derived by Bona, Saut and Toland for describingsmall-amplitude long waves in a water channel, areformally equivalent to the classical Boussinesq systemand correct through first order with regard to asmall parameter characterizing the typicalamplitude-todepth ratio. Exact solutions for a largeclass of systems are presented. The existence of theexact traveling-wave solutions is in general extremely helpful inthe theoretical and numerical study of thesystems.

[1]  J. Fenton.  A ninth-order solution for the solitary wave , 1972, Journal of Fluid Mechanics.

[2]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[3]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[4]  Jerry L. Bona,et al.  A model for the two-way propagation of water waves in a channel , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  A. Newell.  Finite Amplitude Instabilities of Partial Difference Equations , 1977 .

[6]  An Exact Solution of the Wave Equation ut + uux -u(5x) =0 , 1981 .

[7]  J. F. Toland,et al.  Solitary wave solutions for a model of the two-way propagation of water waves in a channel , 1981, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  E. V. Krishnan,et al.  AN EXACT SOLUTION OF THE CLASSICAL BOUSSINESQ EQUATION , 1982 .

[9]  Walter Craig,et al.  An existence theory for water waves and the boussinesq and korteweg-devries scaling limits , 1985 .

[10]  Alan C. Newell,et al.  Solitons in mathematics and physics , 1987 .

[11]  R. Sachs,et al.  On the integrable variant of the Boussinesq system: Painleve´ property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy , 1988 .

[12]  P. Olver,et al.  Existence and Nonexistence of Solitary Wave Solutions to Higher-Order Model Evolution Equations , 1992 .

[13]  R. Dunlap,et al.  Exact traveling wave solutions to nonlinear diffusion and wave equations , 1994 .

[14]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[15]  Mingliang Wang,et al.  Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics , 1996 .

[16]  Jerry L. Bona,et al.  A Boussinesq system for two-way propagation of nonlinear dispersive waves , 1998 .

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