Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces
Bäcklund transformations are used in the calculation of soliton solutions of certain nonlinear evolution equations (NLEEs) [see \textit{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 [\textit{E. J. Parkes} and \textit{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.
scientific article; zbMATH DE number 5015141
scientific article