For the exponential traveling wave solutions to the Hirota bilinear equations, a sufficient and necessary criterion for the existence of linear superposition principle has been given. Motivated by this criterion, we propose a new Hirota bilinear equation via using a multivariate polynomial. Applying the linear superposition principle to this new Hirota bilinear equation, we finally find two types of resonant multiple wave solutions, among which, the resonant three-wave solutions are illustrated with three-dimensional plots.

Resonant behavior of multiple wave solutions to a Hirota bilinear equation

In this paper, the soliton solutions to the quantum Zakharov–Kuznetsov equation in quantum dusty plasmas have been investigated A few integration tools have been applied to extract the soliton solutions to the governing equation The topological, non-topological, singular soliton and complexiton solutions are obtained Additionally, the periodic and singular periodic solutions are listed Finally, the ansatz method has been employed to retrieve a singular 1-soliton solution to the equation for power law nonlinearity In this case, a constraint condition naturally falls out and it serves as a pre-conditioning criteria for singular soliton to exist

Solitons and other solutions to quantum Zakharov–Kuznetsov equation in quantum magneto-plasmas

Abstract In this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

A note on exp-function method combined with complex transform method applied to fractional differential equations

This work studies the solitons and conserved quantities to Biswas–Milovic equation that models the propagation of waves in physical sciences and engineering. Two types of nonlinearity that are Kerr law and parabolic law are taken into consideration. Via the Jacobian elliptic equations expansion approach and \(\psi ^{6}\) model expansion scheme, analytical soliton and soliton-like solutions are derived.

Optical solitons for Biswas–Milovic model with Kerr law and parabolic law nonlinearities

We explore the shape changing and clevaging nature of anti-kink solutions of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. We achieved this by invoking the multiple exp-function method aided with symbolic computation which remains an indispensable tool to deal with computational algebraic systems.

Exact propagating multi-anti-kink soliton solutions of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, using the exp-function method we obtain some new exact solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt (KK) equations. We show figures of some of the new solutions obtained here. We conclude that the exp-function method presents a wider applicability for handling nonlinear partial differential equations.

M. Javidi

Papers

New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup–Kupershmidt Equations