Solitary waves for the generalized equal width (GEW) equation. (English) Zbl 1064.65114

Summary: The authors consider solitary wave solutions of the generalized equal width (GEW) wave equation \(u_t+\varepsilon u^p u_x-\delta u_{xxt}= 0\). This paper presents a collocation method for the GEW equation, which is classified as a nonlinear partial differential equation using quadratic B-splines at midpoints as element shape functions. In this research, the scheme of the equation under investigation is found to be unconditionally stable.
Test problems including the single soliton and the interaction of solitons are used to validate the suggested methods that is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76B25 Solitary waves for incompressible inviscid fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
35Q51 Soliton equations
Full Text: DOI


[1] DOI: 10.1016/S0096-3003(03)00189-9 · Zbl 1038.65101
[2] DOI: 10.1016/S0960-0779(02)00569-6 · Zbl 1030.35139
[3] DOI: 10.1006/jcph.1994.1113 · Zbl 0806.65121
[4] Raslan KR, Al-Azhar University (1999)
[5] DOI: 10.1080/00207160310001614963 · Zbl 1047.65086
[6] DOI: 10.1016/S0045-7825(99)00312-6 · Zbl 0963.76057
[7] Zaki SI, Computational Methods in Applied Mechanics and Engineering 190 pp 4881– · Zbl 1011.76048
[8] DOI: 10.1017/S0022112066001678
[9] DOI: 10.1017/S0022112067002605 · Zbl 0163.21105
[10] DOI: 10.1098/rsta.1972.0032 · Zbl 0229.35013
[11] Bona JL, Fluid Dynamics in Astrophysics and Geophysics. pp pp. 235–267– (1983)
[12] Bona JL, Philosophical Transactions of the Royal Society of London 302 pp 457– (1981)
[13] DOI: 10.1016/0021-9991(92)90054-3 · Zbl 0759.65086
[14] DOI: 10.1016/S0010-4655(99)00471-3 · Zbl 0951.65098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.