Opportunity of representation of the nonlinear wave equations through a variable action-angle

Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential equation is known. Using the known solution of the non-linear differential equation, linear differential equations are set up. The solutions of these linear differential equations are found using standard techniques. Then the solutions of t...

The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able to open a new way in order to calculate or approximate the solution of an ordinary differential equation. Some examples are presented.

In this letter, we study the purely nonlinear oscillator by the method of action-angle variables of Hamiltonian systems. The frequency of the non-isochronous system is obtained, which agrees well with the previously known result. Exact analytic solutions of the system involving generalized trigonometric functions are presented. We also present arguments to show the adiabatic invariance of the action variable for a time-dependent purely nonlinear oscillator.

wave solutions to nonlinear partial differential equations. We simplify the so called (G'/G)-expansion method and apply two of those methods to simple physical problems.

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordina...

A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable potential has also been constructed taking recourse to the above method.

In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.

The problem of group classification of one class of quasilinear equations of hyperbolic type with two independent variables has been solved completely.

In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this non-linear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows, etc. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely $\left(\frac{G'}{G'+G+A}\right)$--expansion method. Using the aforementioned method and computational software, we ha...

Course Notes on Nonlinear Partial Differential Equations of Elliptic Type given to Master students at the University of Craiova, Romania.