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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 \[ u(\xi)=\sum_{i=1}^mw^{i-1}\left[A_iw+B_i\sqrt{\text{sgn}(b) (b+w^2)}\right]+A_0+\sum_{j=0}^nD_j\xi^j. \] On page 104, there are two misprints in equation (21), which should be changed to \[ \begin{cases} u=A_0+A_1w+B_1\sqrt{\text{sgn}(b)(b+w^2)},\\ v=a_0+a_1w+b_1\sqrt{\text{sgn}(b)(b+w^2)}.\end{cases} \] In the paragraph under equation (21), \(\sqrt{1+\text{sgn}(b)w^2}\) should be changed to \(\sqrt{\text{sgn}(b)(b+w^2)}\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35Q51 Soliton equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0987.37064
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References:

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