New Applications to Solitary Wave Ansatz

In this article, the solitary wave and shock wave solitons for nonlinear Ostrovsky equation and Potential Kadomstev-Petviashvili equations have been obtained. The solitary wave ansatz is used to carry out the solutions.


Introduction
Nonlinear wave phenomena appear in various scientific and engineering fields such as electrochemistry, electromagnetics, fluid dynamics, acoustics, cosmology, astrophysics and plasma physics.See references [1]- [4].
In recent time, the numerous approaches have been developed to obtain the solutions of nonlinear equations.
Nonlinear wave is one of the fundamental objects of nature and a growing interest has been given to the propagation of nonlinear waves in the dynamical system.The solitary wave ansatz method [13] [14] is rather heuristic and processes significant features that make it practical for the determination of single soliton solutions for a wide class of nonlinear evolution equations.The solitary wave and shock wave solitons have been obtained, using solitary wave ansatz method, for nonlinear Ostrovsky equation and Potential Kadomstev-Petviashvili (PKP) equation, and we clearly see the consistency, which has recently been applied successfully.
The Ostrovsky equation is, a model of ocean currents motion, read as where β and γ are constants.Parameter β determines the type of dispersion, namely, 1 β = − (negative- dispersion) for surface and internal waves in the ocean and surface waves in a shallow channel with an uneven bottom; 1 β = (positive dispersion) for capillary waves on the surface of liquid or for oblique magneto-acous- tic waves.Parameter > 0 γ measures the effect of rotation.The Potential Kadomstev-Petviashvili (PKP) equation has been considered in the following manner ( ) ( )

Solitary Waves Solitons
In this section, the solitary wave solution or non-topological solution to the Ostrovsky Equation (1.1) and Potential Kadomstev-Petviashvili Equation (1.2) have been found using the following solitary wave ansatz.For this, we have where A is the amplitude of the solitons, B is the inverse width of the solitons and ν is the velocity of the solitary wave.

OS-BBM Equation
From the Equation (2.3), it can be followed tanh cosh After substituting Equations (2.4)-(2.8)into (1.1), the following equation is obtained Solving the above system of equations and also set 1 p = , then it can be written ( ) Hence, the solitary wave solution of the OS-BBM equation is given by ( ) It may be noted that 2 p = is being calculated when exponents 3 p and 2 2 p + are equated equal to each other.Furthermore, set the coefficients of the linearly independent terms to zero.Thus, we can write ( )( )

Shock Waves Solitons
In this section, the shock wave solution or topological solution to the Ostrovsky Equation (1.1) and Potential Kadomstev-Petviashvili Equation (1.2) have been found using the following solitary wave ansatz.For this, we can write

( ) ( )
, tanh , where and 0 where A and B are free parameters and are the amplitude and inverse width of the soliton, while ν is the velocity of the soliton.The value of the exponent p is determined later.

OS-BBM Equation
Following Equation (3.16), it can be written ) Hence, the solitary wave solution of the OS-BBM equation is given by

Potential Kadomstev-Petviashvili (PKP) Equation
From Equation (3.16), it can be followed { }  It may be noted that 2 p = is being calculated when exponents 3 1 p + and 2 3 p + are to be set equal to each other.Furthermore, set the coefficients of the linearly independent terms to zero.It can, thus, be written as )( ) Solving the above system of equations and also set 2 p = , then it can be written solitary wave solution of the Potential Kadomstev-Petviashvili (PKP) equation is given by Furthermore, set the coefficients of the linearly independent terms to zero.Thus, we can write