Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case

Year 2020, Volume: 3 Issue: 2, 44 - 52, 22.06.2020

Pierre Gaillard

https://doi.org/10.32323/ujma.644837

Abstract

From elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these para\-meters goes to $0$, we get rational solutions as a quotient of a polynomial of degree $N(N+1)-2$ in $x$ and $t$, by a polynomial of degree $N(N+1)$ in $x$ and $t$ for each positive integer $N$ depending on $3N$ real parameters. We restrict ourself to give the explicit expressions of these rational solutions for $N=1$ until $N=3$ to shortened the paper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from $1$ to $3$.

Keywords

Boussinesq equation, determinants, Lax pairs, rational solutions

References

• [1] J. Boussinesq, Theorie de l’intumescence appel´ee onde solitaire ou de translation se propageant dans un canal rectangulaire, C.R.A.S., V 72 (1871), 755-759
• [2] J. Boussinesq, Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., Ser. 2, T. 17, (1872), 55-108
• [3] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, L.M.S. Lect. Notes Math., (1991) 149
• [4] P. Deift, C. Tomei, E. Trubowitz, Inverse scattering and the Boussinesq equation, Com. Pure Appl. Math., 35, (1992), 567-628
• [5] M. Toda,Studies of a nonlinear lattice, Phys. Rep., 8, (1975), 1-125
• [6] V.E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillations, Sov. Phys. JETP, 38, (1974), 108-110
• [7] E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, C.U.P., 1990
• [8] R. Hirota, J. Satsuma, Non linear evolution equations generated from the Backlund transformation fot the Boussinesq equation, Prog. of Theor. Phys., 57, (1977), 797-807
• [9] M.J. Ablowitz, J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys., 19, (1978), 2180-2186
• [10] J.J.C. Nimmo, N.C. Freemann, A method of obtaining the N soliton solution of the Boussinesq equation in terms of a wronskian, Phys. Lett., 95, N. 1, (1983), 4-6
• [11] V.B. Matveev, A.O. Smirnov, On the Riemann theta function of a trigonal curve and solutions of the Boussinesq anf KP equations, L.M.P., 14, (1987), 25-31
• [12] V.B. Matveev and M.A. Salle, Darboux transformations and solitons, Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991
• [13] L.V. Bogdanov, V.E. Zakharov, The Boussinesq equation revisited, Phys. D, 165, (2002), 137-162
• [14] P.A. Clarkson, Rational solutions of the Boussinesq equation, Anal. Appl., 6, (2006), 349-369
• [15] P.A. Clarkson, Rational solutions of the classical Boussinesq system, Nonlin. Anal. : Real World Appl., 10, (2010), 3361-3371
• [16] P.A. Clarkson, E. Dowie, Rational solutions of the Boussinesq equation and applications to rogue waves, Trans. of Math. and its Appl., 1, (2017), 1-26