2D solitons in Boussinesq equation with dissipation
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2D solitons in Boussinesq equation with dissipation Kyriakos Christou1 · Marios A. Christou2
Received: 31 March 2015 / Accepted: 1 May 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract We investigate numerically the stationary solutions of 2D Boussinesq type wave equations with square and cubic nonlinearity. In the model equation dissipation is added and we investigate the physical properties of the problem. To investigate and study the problem we implement the Christov spectral technique in L 2 (−∞, ∞). The method was found to be accurate and computationally efficient in other works of the author, mainly in 1D problems. Keywords
Spectral method · Solitons · Boussinesq equations
Mathematics Subject Classification
33F05 · 35Q51 · 35Q53
1 Introduction One of the most important features of generalized wave equations containing nonlinearity and dispersion is that they possess solutions of type of permanent waves which behave in many instances as particles. When the governing system is fully integrable, such waves are called solitons. In 1D a plethora of deep mathematical results have been obtained for solitons (Bullough and Caudrey 1980; Newell 1985; Ablowitz and Sigur 1981). The success was contingent upon the existence of an analytical solution of the respective nonlinear dispersive equation. Naturally, predominant part of the theoretical results were confined to the 1D case. It is of high importance to investigate the 2D case which, in most of the cases, can be done only numerically.
Communicated by Pierangelo Marcati.
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Marios A. Christou [email protected] Kyriakos Christou [email protected]
1
Department of Computer Science and Engineering, European University, Nicosia, Cyprus
2
Department of Mathematics, University of Nicosia, Nicosia, Cyprus
123
K. Christou, M. A. Christou
The first soliton-supporting generalized wave equation (GWE) was derived by Boussinesq (1872) who found its permanent solution to be of sech type. The existence of a localized solution proved that a balance between dispersion and nonlinearity exists. Later on Korteweg and de Vries (1895) derived the evolution equation for the wave amplitude in the moving frame. The same sech is a solution also to KdV equation. To the family of soliton-supporing models that attracted enormous attention in the recent years, one can also add Sine-Gordon and Schroedinger equations. For all these equations, finding 2D solitary wave is a must. One should be able to apply the algorithm developed here for Boussinesq equation to other soliton-supporting equations. Computing a localized solution imposes special requirements on the numerical technique because no boundary conditions are specified at given points, but rather the square of solution is required to be integrable over the infinite domain. Such solution is said to belong to the L 2 (−∞, ∞) space. A number of difficulties are encountered on the way of application of difference and/or finite-element numerical methods to problems in L 2 (−∞, ∞). One of
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