Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion

Nonlinear Dynamics - The licence type in the original article was incorrect and should be CC BY and not CC BY NC.     

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.     

The variational iteration method for exact solutions of Laplace equation

In this work, we introduce a framework for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. Exact solutions are developed by using the He's variational iteration method (VIM). The work confirms the power of the method in reducing the size of calculations.     

Multiple soliton solutions and multiple singular soliton solutions for two integrable systems

Two systems of two-component integrable equations are investigated. The Cole-Hopf transformation and the Hirota's bilinear method are applied for a reliable treatment of these two systems. Multiple-soliton solutions and multiple singular soliton solutions are obtained for each system.     

Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist

In this work we establish two wave modes for the integrable fifth-order Korteweg-de Vries (TfKdV) equations. We determine necessary conditions of the nonlinearity and dispersion parameters of the equation for multiple-soliton solutions to exist. We apply the simplified Hirota method to derive multiple-soliton solutions under these conditions. We also examine the dispersion relations and the phase shifts of the developed models.     

Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator

We develop a variety of negative-order Korteweg-de Vries (KdV) equations in (3+1)-dimensions. The recursion operator of the KdV equation is used to derive these higher dimensional models. The new equations give distinct solitons structures and distinct dispersion relations as well. We also determine multiple soliton solutions for each derived model.