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.   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 (1988), 15.   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 (2000), 603.   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 (1994), 449.   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.   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.   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.   Google Scholar

[8]

C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional series,, in, (1990), 232.   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 (1994), 1095.   Google Scholar

[11]

C. I. Christov, Conservative difference scheme for Boussinesq model of surfacewaves,, in, (1995), 343.   Google Scholar

[12]

C.I. Christov, Numerical investigation of the long-time evolution and interaction of localized waves,, in, (1995), 403.   Google Scholar

[13]

C. I. Christov, G. Maugin, Numerics of some generalized models of lattice dynamics,, in, (1995), 374.   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.   Google Scholar

[15]

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

[16]

C. I. Christov, Solitary waves with Galilean invariance in dispersive shallow-water flows,, in, (2001), 49.   Google Scholar

[17]

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

[18]

G. A. Maugin, C. I. Christov, Nonlinear waves and conservation laws (Nonlnear duality between elastic waves and quasi-particles),, in, (2002), 117.   Google Scholar

[19]

A.C. Newel, "Solitons in Mathematics and Physics,", SIAM, (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, 272 (1972), 47.   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 (1988), 15.   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 (2000), 603.   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 (1994), 449.   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.   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.   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.   Google Scholar

[8]

C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional series,, in, (1990), 232.   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 (1994), 1095.   Google Scholar

[11]

C. I. Christov, Conservative difference scheme for Boussinesq model of surfacewaves,, in, (1995), 343.   Google Scholar

[12]

C.I. Christov, Numerical investigation of the long-time evolution and interaction of localized waves,, in, (1995), 403.   Google Scholar

[13]

C. I. Christov, G. Maugin, Numerics of some generalized models of lattice dynamics,, in, (1995), 374.   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.   Google Scholar

[15]

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

[16]

C. I. Christov, Solitary waves with Galilean invariance in dispersive shallow-water flows,, in, (2001), 49.   Google Scholar

[17]

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

[18]

G. A. Maugin, C. I. Christov, Nonlinear waves and conservation laws (Nonlnear duality between elastic waves and quasi-particles),, in, (2002), 117.   Google Scholar

[19]

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

[1]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471

[2]

Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Nicolas Seguin. An energy-consistent depth-averaged Euler system: Derivation and properties. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 961-988. doi: 10.3934/dcdsb.2015.20.961

[3]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002

[4]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801

[5]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837

[6]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47

[7]

H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119

[8]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101

[9]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91

[10]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857

[11]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897

[12]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703

[13]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003

[14]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036

[15]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465

[16]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767

[17]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509

[18]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675

[19]

Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557

[20]

Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112

 Impact Factor: 

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]