The Soliton-Like Solutions of ϕ ⁶ and ϕ ⁴ + ϕ ³ Models with and Without Dissipation

Abstract

Besides the <f>* theory, the <f> 6 model and the <f>* + <f> 3 model are aiso useful in physics. Some specific types of solutions of them can be obtained from the solutions of <j>* model. The important solutions previously known are the special examples of ours. The dissipation equations of these models are also studied. The results indicate that the friction effects spread the singular solutions with velocity v = 1 (Heaviside function) of the undamped equations such that they are nonsingular and their velocities are lower than one.

New Types of Exact Solutions for (N+1)-Dimensional ϕ 4 Model

“…Because of the existence of the deformation relations among the φ 4 -model and the other nonlinear Klein-Gordon systems such as the φ 6 -equation, T.D. Lee model, multiple sine-Gordon systems, etc., [4,5,9] the new type solutions of φ 4 proposed in this paper will lead to new types of other types of nonlinear Klein-Gordon equations.…”

“…Though the solution is known in Ref. [5], one can still find more general solution by including an arbitrary function to the "independent" variable V , say,…”

“…Some properties and exact solutions for the φ 4 field are known in Refs. [4] and [5]. However, there are still many open important problems.…”

“…Though many special relations of the φ 4 model have been given in Ref. [5], many kinds of exact solutions such as the periodic-periodic interaction solutions have not yet been found. Here we give out some new exact solutions of the φ 4 -model by extending the mapping relations of Ref.…”

Jacobi Elliptic Function Solutions of Some Nonlinear Evolution Equations

“…Because of the availability of symbolic computation (Maple, Mathematica or Matlab), various effective methods have been proposed, such as symmetry reduction method, [6,7] Hirota bilinear method, [4,8] and function expansion method. [9,10] Usually, the one-soliton solutions of NLEEs are special limited cases of the periodic function solutions that are expressed by Jacobi elliptic functions. In this letter, we obtain some new exact solutions which include Jacobi elliptic function solutions and corresponding soliton-like solutions of the M -coupled KdV equation, M -coupled Ito equation, vKdV equation, and AKNS equation.…”