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THE DYNAMIC SYSTEM THEORY OF A NEW FORM OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
  • Esen Hanaç
Esen Hanaç
Adiyaman University Faculty of Arts and Sciences

Corresponding Author:[email protected]

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Abstract

We examine the main results concerning the existence and structure of travelling waves (TWs) which may occur in the following equation namely, u_t+uu_x=u_xx+u(1-u)(u-m),-∞0, where ≠ 0, represents distance, represents time and > 2 is a constant. We use dynamical system theory to obtain the results of the equation and exhibit a phase-space of their stable points. As a beginning we start to get ordinary differential equation form of above equation after substituting of a new transformation into it. Additionally, all points are indicated depending on the structure of eigenvalues of the critical points in phase-space by using a generated matlab implementation of ode45 package. Our goal is to find a heteroclinic trajectory from unstable node to stable node in parallel with travelling wave solutions for the minimum waving accelerate and the structure of the other soliton solutions to be defined. Furthermore the good agreement of the convergence of exact solution of above equation and numerical solution of dynamic system of the equation found by applying parabolic method, is demonstrated. The results of our study demonstrate that the equation given above can confirm soliton solutions.