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Two integrable coupled nonlinear systems. (English) Zbl 0944.35077
Summary: Motivated by Hirota and Satsuma’s results on their coupled KdV equation, two integrable coupled nonlinear systems are considered. One of them is a coupled Ito system. It is shown that the coupled Ito system is a special case of the $\left(6,2\right)$-reduction of the two component BKP hierarchy while the other coupled system can be obtained from the $\left(5,1\right)$-reduction of the two component BKP hierarchy. By using MATHEMATICA, we obtain the 3- and 4-soliton solutions of the coupled Ito system. In addition, starting from bilinear equations of the other coupled system, a Bäcklund transformation is found and nonlinear superposition formulae are established. Soliton solutions and rational solutions are also derived from these results.
MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 37-04 Machine computation, programs (Dynamical systems and ergodic theory)
Mathematica