# American Institute of Mathematical Sciences

2013, 2013(special): 139-148. doi: 10.3934/proc.2013.2013.139

## Investigation of the long-time evolution of localized solutions of a dispersive wave system

 1 Dept. of Mathematics, University of Louisiana at Lafayette, LA, United States 2 Dept. of Applied Mathematics and Informatics, Technical University of Sofia, 1000 Sofia, Bulgaria

Received  October 2012 Published  November 2013

We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.
Citation: C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139
##### References:
 [1] T. B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil. Trans. R. Soc. A 272, 272 (1972), 47. [2] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation,, Commun. Math. Phys., 118 (1988), 15. [3] J. L. Bona, W. R. McKinney and J. M. Restrepo, Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation,, J. Nonlinear Sci., 10 (2000), 603. [4] J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves,, J. Nonlinear Sci., 4 (1994), 449. [5] J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire,, Comp. Rend. Hebd. des Seances de l'Acad. des Sci., 72 (1871), 755. [6] J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal,, Comp. Rend. Hebd. des Seances de l'Acad. des Sci., 73 (1871), 256. [7] J. V. Boussinesq, Théorie 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,, Journal de Mathématiques Pures et Appliquées, 17 (1872), 55. [8] C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional series,, in, (1990), 232. [9] C. I. Christov, "Gaussian elimination with pivoting for multidiagonal systems'',, Internal Report 4. University of Reading, (1994). [10] C. I. Christov, and M. G. Velarde, Inelastic interaction of Boussinesq solitons,, Int. J. Bifurcation and Chaos, 4 (1994), 1095. [11] C. I. Christov, Conservative difference scheme for Boussinesq model of surfacewaves,, in, (1995), 343. [12] C.I. Christov, Numerical investigation of the long-time evolution and interaction of localized waves,, in, (1995), 403. [13] C. I. Christov, G. Maugin, Numerics of some generalized models of lattice dynamics,, in, (1995), 374. [14] C. I. Christov, G. A. Maugin, An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system,, J. Comp. Phys., 116 (1995), 39. [15] C. I. Christov, An energy-consistent dispersive shallow-water model,, Wave Motion, 4 (2001), 161. [16] C. I. Christov, Solitary waves with Galilean invariance in dispersive shallow-water flows,, in, (2001), 49. [17] Y. Liu, Instability of solitary waves for generalized Boussinesq equations,, J. Dyn. Diff. Eq., 5 (1993), 537. [18] G. A. Maugin, C. I. Christov, Nonlinear waves and conservation laws (Nonlnear duality between elastic waves and quasi-particles),, in, (2002), 117. [19] A.C. Newel, "Solitons in Mathematics and Physics,", SIAM, (1985).

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##### References:
 [1] T. B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil. Trans. R. Soc. A 272, 272 (1972), 47. [2] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation,, Commun. Math. Phys., 118 (1988), 15. [3] J. L. Bona, W. R. McKinney and J. M. Restrepo, Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation,, J. Nonlinear Sci., 10 (2000), 603. [4] J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves,, J. Nonlinear Sci., 4 (1994), 449. [5] J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire,, Comp. Rend. Hebd. des Seances de l'Acad. des Sci., 72 (1871), 755. [6] J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal,, Comp. Rend. Hebd. des Seances de l'Acad. des Sci., 73 (1871), 256. [7] J. V. Boussinesq, Théorie 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,, Journal de Mathématiques Pures et Appliquées, 17 (1872), 55. [8] C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional series,, in, (1990), 232. [9] C. I. Christov, "Gaussian elimination with pivoting for multidiagonal systems'',, Internal Report 4. University of Reading, (1994). [10] C. I. Christov, and M. G. Velarde, Inelastic interaction of Boussinesq solitons,, Int. J. Bifurcation and Chaos, 4 (1994), 1095. [11] C. I. Christov, Conservative difference scheme for Boussinesq model of surfacewaves,, in, (1995), 343. [12] C.I. Christov, Numerical investigation of the long-time evolution and interaction of localized waves,, in, (1995), 403. [13] C. I. Christov, G. Maugin, Numerics of some generalized models of lattice dynamics,, in, (1995), 374. [14] C. I. Christov, G. A. Maugin, An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system,, J. Comp. Phys., 116 (1995), 39. [15] C. I. Christov, An energy-consistent dispersive shallow-water model,, Wave Motion, 4 (2001), 161. [16] C. I. Christov, Solitary waves with Galilean invariance in dispersive shallow-water flows,, in, (2001), 49. [17] Y. Liu, Instability of solitary waves for generalized Boussinesq equations,, J. Dyn. Diff. Eq., 5 (1993), 537. [18] G. A. Maugin, C. I. Christov, Nonlinear waves and conservation laws (Nonlnear duality between elastic waves and quasi-particles),, in, (2002), 117. [19] A.C. Newel, "Solitons in Mathematics and Physics,", SIAM, (1985).
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