Elliptic Function Periodic Solutions Research Articles

Exact single travelling wave solutions to the fractional perturbed Gerdjikov–Ivanov equation in nonlinear optics

In this paper, exact single travelling wave solutions to the nonlinear fractional perturbed Gerdjikov–Ivanov equation are captured by the complete discrimination system for polynomial method and the trial equation method. In the classification, we can find out the original equation has rational function solutions, solitary wave solutions, triangular function periodic solutions, and elliptic function periodic solutions, which are normally very difficult to be obtained by other methods. In particular, the concrete parameters are set to show that the solutions in the classification can be realized in almost all cases.

Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity

In this Letter, we investigate the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.

On Exact Traveling Wave Solutions for (1 + 1) Dimensional Kaup-Kupershmidt Equation

In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods.

New explicit solutions for the Klein–Gordon equation with cubic nonlinearity

In this paper, we investigate Klein–Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.

New explicit exact solutions of one type of generalized Boussinesq equations and the Boussinesq-Burgers equation

In this paper, explicit exact solutions of one type of generalized Boussinesq equations (B(m，n) equations) and the Boussinesq-Burgers equation (B-B equation) are investigated using a generalized algebraic method. A variety of explicit exact solutions, such as solitary wave, triangular periodic, rational, Jacobi and Weierstrass elliptic function periodic solutions are obtained.

A series of explicit and exact travelling wave solutions of the B( m, n) equations

In this paper, explicit and exact travelling wave solutions of B( m, n) equations are investigated by using a generalized algebraic method. A series of explicit exact solutions, such as solitary wave, triangular periodic, rational, Jacobi and Weierstrass elliptic function periodic solutions are obtained. Many new results are presented.

Trial equation method and its applications to nonlinear evolution equations

A new method, that is, trial equation method, was given to obtain the exact trav eling wave solutions for nonlinear evolution equations. As an example, a class o f fifth-order nonlinear evolution equations was discussed. Its exact traveling w ave solutions, which included rational form solutions, solitary wave solutions, triangle function periodic solutions, polynomial type Jacobian elliptic function periodic solutions and fractional type Jacobian elliptic function periodic solu tions, were given.

New periodic solutions of Ito’s 5th-order mKdV equation and Ito’s 7th-order mKdV equation

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh-function method, a new algebraic method is presented to obtain multiple travelling wave solutions for nonlinear wave equations. By using the method, Ito’s 5th-order and 7th-order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found. With modulus m → 1 or m → 0, these solutions degenerate into corresponding solitary wave solutions, shock wave solutions and trigonometric function solutions.