Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.

The integrability of the coupled Ramani equation with binary Bell polynomials

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.

Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations

Under investigation in this paper are the coupled Ablowitz–Ladik equations, which are linked to the optical fibres, waveguide arrays, and optical lattices. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation. Bright/dark one- and two-soliton solutions are also obtained. Asymptotic analysis indicates that the interactions between the bright/dark two solitons are elastic. Amplitudes and velocities of the bright solitons increase as the value of the lattice spacing increases. Increasing value of the lattice spacing can lead to the increase of both the bright solitons’ amplitudes and velocities, and the decrease of the velocities of the dark solitons. The lattice spacing parameter has no effect on the amplitudes of the dark solitons. Overtaking interaction between the unidirectional bright two solitons and a bound state of the two equal-velocity solitons is presented. Overtaking interaction between the unidirectional dark two solitons and the two parallel dark solitons is also plotted.

Solitons for a (2+1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko equation in a fluid

In this letter, a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko equation is investigated, which describes the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Under two different constraints of the time-dependent coefficients in this equation, two different bilinear forms are derived by virtue of the binary Bell polynomials. Multiple solitary waves are constructed via the Hirota method, whose propagation properties and interaction characteristics are investigated graphically as well. Propagation and interaction of the solitons are illustrated graphically: (i) time-dependent coefficients affect the shape of the solitons; (ii) interaction of the solitons is elastic, i.e., amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

Solitons and integrability for a (2+1)-dimensional generalized variable-coefficient shallow water wave equation

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg–de Vries (KdV) equation and the Calogero–Bogoyavlenskii–Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

Multi-soliton Collisions and Bäcklund Transformations for the (2+1)-dimensional Modified Nizhnik–Novikov–Vesselov Equations

The Korteweg–de Vries (KdV)-type equations can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, while the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations are the isotropic extensions of KdV-type equations. In this paper, we investigate the (2+1)-dimensional modified Nizhnik–Novikov–Vesselov equations. By virtue of the binary Bell polynomials, bilinear forms, multi-soliton solutions and Bäcklund transformations are derived. Effects of some parameters on the solitons and monotonic function are graphically illustrated. We can observe the coalescence of the two solitons in their collision region, where their shapes change after the collision.

Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, theN-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.