American Institute of Mathematical Sciences

Advanced
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

C. I. Christov 1, and  M. D. Todorov 2,

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.
Keywords: self-similar solutions., long-time evolution, dispersive wave system, Energy-consistent Boussinesq paradigm.
Mathematics Subject Classification: Primary: 65M06, 35Q53; Secondary: 35Q5.
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, (1972), 47-78.  Google Scholar

[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(1) (1988), 15-29.  Google Scholar

[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(6) (2000), 603-638.  Google Scholar

[4]

J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves, J. Nonlinear Sci., 4(1) (1994), 449-470.  Google Scholar

[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), 755-759. Google Scholar

[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), 256-260. Google Scholar

[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), 55-108. Google Scholar

[8]

C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional series, in "Continuum Models and Discrete Systems'', (ed. G. A. Maugin), Longman, (1990), vol. 1, 232-253.  Google Scholar

[9]

C. I. Christov, "Gaussian elimination with pivoting for multidiagonal systems'', Internal Report 4. University of Reading, 1994. Google Scholar

[10]

C. I. Christov, and M. G. Velarde, Inelastic interaction of Boussinesq solitons, Int. J. Bifurcation and Chaos, 4(5) (1994), 1095-1112. Google Scholar

[11]

C. I. Christov, Conservative difference scheme for Boussinesq model of surfacewaves, in "Proc. ICFD V'' (eds. W.K. Morton and M.J. Baines), Oxford University Press, (1995), 343-349. Google Scholar

[12]

C.I. Christov, Numerical investigation of the long-time evolution and interaction of localized waves, in "Fluid Physics, Proceedings of the Summer Schools'' (eds. M.G. Velarde, C.I. Christov), World Scientific, Singapore, (1995), 403-422. Google Scholar

[13]

C. I. Christov, G. Maugin, Numerics of some generalized models of lattice dynamics, in "Nonlinear Waves in Solids'', (eds. J. L. Wegner and F. R. Norwood), ASME Book No AMR 137, (1995), 374-379. Google Scholar

[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), 39-51.  Google Scholar

[15]

C. I. Christov, An energy-consistent dispersive shallow-water model, Wave Motion, 4(5) (2001), 161-174.  Google Scholar

[16]

C. I. Christov, Solitary waves with Galilean invariance in dispersive shallow-water flows, in "Differential Equations and Nonlinear Mechanics'', (ed. K. Vajravelu), Kluwer (2001), 49-68.  Google Scholar

[17]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dyn. Diff. Eq., 5(3) (1993), 537-558.  Google Scholar

[18]

G. A. Maugin, C. I. Christov, Nonlinear waves and conservation laws (Nonlnear duality between elastic waves and quasi-particles), in "Topics in Nonlinear Wave Mechanics'', (eds. C. I. Christov & A. Guran), Birkhäuser, Boston (2002), 117-160.  Google Scholar

[19]

A.C. Newel, "Solitons in Mathematics and Physics," SIAM, Philadelphia, 1985. Google Scholar

show all references

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, (1972), 47-78.  Google Scholar

[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(1) (1988), 15-29.  Google Scholar

[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(6) (2000), 603-638.  Google Scholar

[4]

J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves, J. Nonlinear Sci., 4(1) (1994), 449-470.  Google Scholar

[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), 755-759. Google Scholar

[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), 256-260. Google Scholar

[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), 55-108. Google Scholar

[8]

C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional series, in "Continuum Models and Discrete Systems'', (ed. G. A. Maugin), Longman, (1990), vol. 1, 232-253.  Google Scholar

[9]

C. I. Christov, "Gaussian elimination with pivoting for multidiagonal systems'', Internal Report 4. University of Reading, 1994. Google Scholar

[10]

C. I. Christov, and M. G. Velarde, Inelastic interaction of Boussinesq solitons, Int. J. Bifurcation and Chaos, 4(5) (1994), 1095-1112. Google Scholar

[11]

C. I. Christov, Conservative difference scheme for Boussinesq model of surfacewaves, in "Proc. ICFD V'' (eds. W.K. Morton and M.J. Baines), Oxford University Press, (1995), 343-349. Google Scholar

[12]

C.I. Christov, Numerical investigation of the long-time evolution and interaction of localized waves, in "Fluid Physics, Proceedings of the Summer Schools'' (eds. M.G. Velarde, C.I. Christov), World Scientific, Singapore, (1995), 403-422. Google Scholar

[13]

C. I. Christov, G. Maugin, Numerics of some generalized models of lattice dynamics, in "Nonlinear Waves in Solids'', (eds. J. L. Wegner and F. R. Norwood), ASME Book No AMR 137, (1995), 374-379. Google Scholar

[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), 39-51.  Google Scholar

[15]

C. I. Christov, An energy-consistent dispersive shallow-water model, Wave Motion, 4(5) (2001), 161-174.  Google Scholar

[16]

C. I. Christov, Solitary waves with Galilean invariance in dispersive shallow-water flows, in "Differential Equations and Nonlinear Mechanics'', (ed. K. Vajravelu), Kluwer (2001), 49-68.  Google Scholar

[17]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dyn. Diff. Eq., 5(3) (1993), 537-558.  Google Scholar

[18]

G. A. Maugin, C. I. Christov, Nonlinear waves and conservation laws (Nonlnear duality between elastic waves and quasi-particles), in "Topics in Nonlinear Wave Mechanics'', (eds. C. I. Christov & A. Guran), Birkhäuser, Boston (2002), 117-160.  Google Scholar

[19]

A.C. Newel, "Solitons in Mathematics and Physics," SIAM, Philadelphia, 1985. Google Scholar

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