Shikuo Liu

JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS

Abstract A Jacobi elliptic function expansion method, which is more general than the hyperbolic tangent function expansion method, is proposed to construct the exact periodic solutions of nonlinear wave equations. It is shown that the periodic solutions obtained by this method include some shock wave solutions and solitary wave solutions.

New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations

Abstract New Jacobi elliptic functions are applied in Jacobi elliptic function expansion method to construct the exact periodic solutions of nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this method and more shock wave solutions or solitary wave solutions can be got at their limit condition.

New transformations and new approach to find exact solutions to nonlinear equations

New transformations from the nonlinear sine-Gordon equation are shown in this Letter, based on them a new approach is proposed to construct exact periodic solutions to nonlinear equations. It is shown that more new periodic solutions can be obtained by this new approach and more shock wave solutions or solitary wave solutions can be got under their limit condition.

New kinds of solutions to Gardner equation

Abstract On the basis of analysis to the projective Riccati equations, an intermediate transformation in expansion method is constructed. And this transformation is applied to solve Gardner equation, there many new kinds of travelling wave solutions including solitary wave solution are obtained, in which some are found for the first time.

The JEFE method and periodic solutions of two kinds of nonlinear wave equations

Abstract The Jacobi elliptic function expansion (JEFE) method is applied to construct the exact periodic solutions to two kinds of nonlinear wave equations, such as BBM equation, fifth-order dispersive equation, Kawahara equation, modified Kawahara equation, second-order BO equation and symmetrical-regular long wave equation. The corresponding shock wave solutions or solitary wave solutions are obtained as special cases of the periodic solutions. It is shown that this method is very powerful for some nonlinear wave equations, and its applying domain is given.

Exact Solutions to Double and Triple Sinh-Gordon Equations

In this paper, two transformations are introduced to solve double sinh-Gordon equation and triple sinh-Gordon equation, respectively. It is shown that different transformations are required in order to obtain more kinds of solutions to different types of sinh-Gordon equations. - PACS: 03.65.Ge

Novel exact solutions to the short pulse equation

In this paper, the bridge connecting the short pulse equation (SPE for short) with the sine-Gordon equation is applied to construct the novel solutions to the short pulse equation. It is shown that the solutions of the sine-Gordon equation can be used to obtain many different kinds of solutions to the short pulse equation with the aid of symbolic computation and plot representation of Maple.

Breather solutions and breather lattice solutions to the sine-Gordon equation

In this paper, dependent and independent variable transformations are introduced to solve the sine–Gordon (SG) equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained for the (SG) equation, including breather solutions and breather lattice solutions.

The Hopf bifurcation and spiral wave solution of the complex Ginzburg–Landau equation

Abstract When the dispersive and diffusive effects are negligible, the complex Ginzburg–Landau equation degenerates, in form, as the Landau equation, in which occurs the Hopf bifurcation. For the traveling wave solutions of the complex Ginzburg–Landau equation, which is reduced to its corresponding ordinary differential form by use of traveling wave frame, the spiral and plane waves correspond to the orbit near the focus and the limit cycle, respectively. The shock wave is a heteroclinic orbit between the focus and limit cycle.

The most intensive fluctuation in chaotic time series and relativity principle

Relativity principle in mechanics and principle of invariant speed of light lead to Einstein theory. The exponent of porder momentum, derived from a piece of multi-scale chaotic time series, varies with the order p and cannot exceeds a maximum, so there exists the principle of scale relativity. Its special case is the same one as Lorenz transformation from Einstein theory. 2002 Elsevier Science Ltd. All rights reserved.