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New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. (English) Zbl 1092.35524

Summary: In this Letter, a more powerful method to seek exact travelling wave solutions of nonlinear partial differential equations is presented, which uses the good ideas of the extended-tanh function method and our previous method. The two new integrable coupled potential KdV equation and modified KdV-type equations which were firstly presented by M. V. Foursov [J. Math. Phys. 41, No. 9, 6173–6185 (2000; Zbl 0987.37064)] are chosen to illustrate the method by using symbolic computation such that multiple travelling wave solutions are obtained which contain new kink-like soliton solutions, kink-shaped solitons, bell-shaped solitons, rational solutions and singular solitons that may be useful to explain the “blow-up” phenomena.

Summary of the erratum: On page 101, there is a misprint in equation (9), which should be changed to

u(ξ)= i=1 m w i-1 A i w+B i sgn(b)(b+w 2 )+A 0 + j=0 n D j ξ j ·

On page 104, there are two misprints in equation (21), which should be changed to

u=A 0 +A 1 w+B 1 sgn(b)(b+w 2 ),v=a 0 +a 1 w+b 1 sgn(b)(b+w 2 )·

In the paragraph under equation (21), 1+sgn(b)w 2 should be changed to sgn(b)(b+w 2 ).

35Q53KdV-like (Korteweg-de Vries) equations
35B40Asymptotic behavior of solutions of PDE
35Q51Soliton-like equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies