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Khater, A. H.; Callebaut, D. K.; Sayed, S. M.

Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces. (English) Zbl 1093.35005

J. Comput. Appl. Math. 189, No. 1-2, 387-411 (2006).
Bäcklund transformations are used in the calculation of soliton solutions of certain nonlinear evolution equations (NLEEs) [see R. K. Dodd, J. Phys. A 11, 81–92 (1978; Zbl 0368.35058)]. For some nonlinear evolution equations which describe pseudo-spherical surfaces two new exact solution classes are generated. Exact travelling wave and solitary wave solutions for Burgers’ equation \(2u_t-2uu_x-u_{xx}=0\) and also for a fifth-order evolution equation \(u_t+au^2u_x+bu_xu_{xx}+ huu_{xxx}+du_{xxxxx} =0\), where \(a,b,h\), and \(d\) are constants, containing the fifth-order Korteweg-de Vries equation and also the Sawada-Kotera equation [E. J. Parkes and B. R. Duffy, Comput. Phys. Commun. 98, No. 3, 288–300 (1996; Zbl 0948.76595)] are obtained by using an improved sine-cosine method and the Wu’s elimination method. Employing the Bäcklund transformations involving explicitly the wave solutions, new solutions are generated for Burgers’ equation. The obtained results are illustrated.
Reviewer: Valery V. Karachik (Chelyabinsk)

Cited in 13 Documents

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
58J90 Applications of PDEs on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53A05 Surfaces in Euclidean and related spaces
35Q53 KdV equations (Korteweg-de Vries equations)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations

Keywords:

Bäcklund transformations; travelling wave; solitary wave; Burgers’ equation; Sawada-Kotera equation

Citations:

Zbl 0368.35058; Zbl 0948.76595

Software:

ATFM
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\textit{A. H. Khater} et al., J. Comput. Appl. Math. 189, No. 1--2, 387--411 (2006; Zbl 1093.35005)
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References:

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