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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

$u\left(\xi \right)=\sum _{i=1}^{m}{w}^{i-1}\left[{A}_{i}w+{B}_{i}\sqrt{\text{sgn}\left(b\right)\left(b+{w}^{2}\right)}\right]+{A}_{0}+\sum _{j=0}^{n}{D}_{j}{\xi }^{j}·$

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

$\left\{\begin{array}{c}u={A}_{0}+{A}_{1}w+{B}_{1}\sqrt{\text{sgn}\left(b\right)\left(b+{w}^{2}\right)},\hfill \\ v={a}_{0}+{a}_{1}w+{b}_{1}\sqrt{\text{sgn}\left(b\right)\left(b+{w}^{2}\right)}·\hfill \end{array}\right\$

In the paragraph under equation (21), $\sqrt{1+\text{sgn}\left(b\right){w}^{2}}$ should be changed to $\sqrt{\text{sgn}\left(b\right)\left(b+{w}^{2}\right)}$.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35B40 Asymptotic behavior of solutions of PDE 35Q51 Soliton-like equations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies