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Bright and dark soliton solutions and Bäcklund transformation for the Eckhaus-Kundu equation with the cubic-quintic nonlinearity. (English) Zbl 1328.37055
Summary: With symbolic computation, the Eckhaus-Kundu equation which appears in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics, is studied via the Hirota method. By virtue of the dependent variable transformation, the bilinear form is obtained. Bilinear Bäcklund transformation is given with the help of exchange formulae and the corresponding one-soliton solution is derived. Bright and dark $$N$$-soliton solutions are obtained. Propagation and interaction of the bright and dark solitons are discussed analytically and graphically. Interactions of the two solitons are presented. Bound state of the two solitons can be suppressed via the choice of parameters.

##### MSC:
 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 35C07 Traveling wave solutions 35Q40 PDEs in connection with quantum mechanics
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