| | | |

## Rational Solutions to the Boussinesq Equation

#### Pierre Gaillard [1]

##### 61 105

Rational solutions to the Boussinesq equation are constructed as a quotient of two polynomials in $x$ and $t$. For each positive integer $N$, the numerator is a polynomial of degree $N(N+1)-2$ in $x$ and $t$, while the denominator is a polynomial of degree $N(N+1)$ in $x$ and $t$. So we obtain a hierarchy of rational solutions depending on an integer $N$ called the order of the solution. We construct explicit expressions of these rational solutions for $N=1$ to $4$.
Boussinesq equation, Rational solutions, Rogue wave
• [1] J. Boussinesq, Theorie de l’intumescence appelee onde solitaire ou de translation se propageant dans un canal rectangulaire, C.R.A.S., 72 (1871), 755179.
• [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 sensiblemant parielles de la surface au fond, J. Math. Pures Appl., 7 (1872), 55178.
• [3] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser., 149 (1991), C.U.P.
• [4] P. Deift, C. Tomei, E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math, 35 (1982), 567178
• [5] M. Toda, Studies of a nonlinear lattice, Phys. Rep., 8 (1975), 1175.
• [6] V. E. Zakharov, On stocastization of one-dimensional chains of nonlinear oscillations, Sov. Phys. JETP, 38 (1974), 108170.
• [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 B¨acklund transformation fot the Boussinesq equation, Prog. of Theor. Phys., 57 (1977), 797177.
• [9] M. J. Ablowitz, J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys., 19 (1978), 21801786.
• [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(1) (1983), 417.
• [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, 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), 137172.
• [14] P. A. Clarkson, Rational solutions of the Boussinesq equation, Anal. Appl., 6 (2008), 349179.
• [15] P. A. Clarkson, Rational solutions of the classical Boussinesq system, Nonlin. Anal. : Real World Appl., 10 (2010), 33611771
• [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), 117.
Primary Language en Mathematics Articles Orcid: 0000-0002-7073-8284Author: Pierre Gaillard (Primary Author)Institution: Université de BourgogneCountry: France Publication Date: June 17, 2019
 Bibtex @research article { fujma512333, journal = {Fundamental Journal of Mathematics and Applications}, issn = {2645-8845}, address = {Murat KİRİŞCİ}, year = {2019}, volume = {2}, pages = {1 - 4}, doi = {10.33401/fujma.512333}, title = {Rational Solutions to the Boussinesq Equation}, key = {cite}, author = {Gaillard, Pierre} } APA Gaillard, P . (2019). Rational Solutions to the Boussinesq Equation. Fundamental Journal of Mathematics and Applications, 2 (1), 1-4. DOI: 10.33401/fujma.512333 MLA Gaillard, P . "Rational Solutions to the Boussinesq Equation". Fundamental Journal of Mathematics and Applications 2 (2019): 1-4 Chicago Gaillard, P . "Rational Solutions to the Boussinesq Equation". Fundamental Journal of Mathematics and Applications 2 (2019): 1-4 RIS TY - JOUR T1 - Rational Solutions to the Boussinesq Equation AU - Pierre Gaillard Y1 - 2019 PY - 2019 N1 - doi: 10.33401/fujma.512333 DO - 10.33401/fujma.512333 T2 - Fundamental Journal of Mathematics and Applications JF - Journal JO - JOR SP - 1 EP - 4 VL - 2 IS - 1 SN - 2645-8845- M3 - doi: 10.33401/fujma.512333 UR - https://doi.org/10.33401/fujma.512333 Y2 - 2019 ER - EndNote %0 Fundamental Journal of Mathematics and Applications Rational Solutions to the Boussinesq Equation %A Pierre Gaillard %T Rational Solutions to the Boussinesq Equation %D 2019 %J Fundamental Journal of Mathematics and Applications %P 2645-8845- %V 2 %N 1 %R doi: 10.33401/fujma.512333 %U 10.33401/fujma.512333 ISNAD Gaillard, Pierre . "Rational Solutions to the Boussinesq Equation". Fundamental Journal of Mathematics and Applications 2 / 1 (June 2019): 1-4. https://doi.org/10.33401/fujma.512333 AMA Gaillard P . Rational Solutions to the Boussinesq Equation. FUJMA. 2019; 2(1): 1-4. Vancouver Gaillard P . Rational Solutions to the Boussinesq Equation. Fundamental Journal of Mathematics and Applications. 2019; 2(1): 4-1.