Whitham-Broer-Kaup (WBK) equations describing the propagation of shallow-water waves, with a variable transformation, are transformed into a generalized Ablowitz-Kaup-Newell-Segur system, the bilinear forms of which are obtained via the rational transformations. Employing the matrix extension and symbolic computation, we derive types of solutions of the WBK equations through the selection of different canonical matrices, including solitons, rational solutions, and complexitons. Furthermore, dynamic properties of the solutions are discussed graphically and a novel phenomenon is observed, i.e., the coexistence of the elastic-inelastic interactions without disturbing each other.

By symbolic computation we study a variable-coefficient derivative nonlinear Schrödinger (vc-DNLS) equation describing nonlinear Alfvén waves in inhomogeneous plasmas. Based on the Lax pair of the vc-DNLS equation, the N-fold Darboux transformation is constructed via a gauge transformation and the reduction technique. Multi-solitonic solutions in terms of the double Wronskian for the vc-DNLS equation are obtained. Two-and three-solitonic interactions are analyzed graphically, i.e., overtaking, head-on and parallel interactions. Plasma streaming and inhomogeneous magnetic field control the amplitudes and velocities of the solitonic waves, respectively. The nonuniform density affects the amplitudes of the solitonic waves. The effects of the spectral parameters on the dynamics of the two-solitonic waves are discussed.Our results might facilitate the analytic investigation on certain inhomogeneous systems in the Earth's magnetosphere, solar winds, planetary bow shocks, dusty cometary tails and interplanetary shocks.

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This paper is to investigate the extended (2+1)-dimensional Konopelchenko-Dubrovsky equations, which can be applied to describing certain phenomena in the stratified shear flow, the internal and shallow-water waves, plasmas and other fields. Painlevé analysis is passed through via symbolic computation. Bilinear-form equations are constructed and soliton solutions are derived. Soliton solutions and interactions are illustrated. Bilinear-form Bäcklund transformation and a type of solutions are obtained.

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