Bäcklund transformation and soliton solutions for two (3+1)-dimensional nonlinear evolution equations

In this paper, two (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equations (NLEEs) are under investigation by employing the Hirota’s method and symbolic computation. We derive the bilinear form and bilinear Bäcklund transformation (BT) for the two NLEEs. Based on the bilinear form, we obtain the multi-soliton solutions for them. Furthermore, multi-soliton solutions in terms of Wronskian determinant for the first NLEE are constructed, whose validity is verified through direct substitution into the bilinear equations.

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.

Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

For the famous Darboux-Pöschl-Teller equation, we present new wronskian representation both for the potential and the related eigenfunctions. The simplest application of this new formula is the explicit description of dynamics of the DPT potentials and the action of the KdV hierarchy. The key point of the proof is some evaluation formulas for special wronskian determinant.