Shida Liu

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.

Preliminary Studies on Flashover Mechanism in Compartment Fires

The mechanism of flashover in a compartment fire is investigated using nonlinear theory through a two-layer zone model. To simplify the illustration of the approach, there is only one variable, i.e. the upper smoke temperature, with the smoke layer interface height being fixed. Effects of different factors, such as the geometry of the compartment, the thermal and chemical properties of the combustible content, and the ventilation conditions, affecting flashover are discussed. Results show that the occurrence of flashover depends on many factors, among which, heat release rate is the most important.

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.

MULTIPLE ATTRACTORS OF MIXED CONVECTION IN CONFINED SPACES

Dynamical structure and physical mechanism of multiple attractors in mixed convection are investigated using a simplified model in the form of ordinary differential equations. Stability analysis and bifurcation analysis on the dynamical behavior show that when the Archimedes number is fixed in some region, an inverted bifurcation takes place as the Reynolds number is increased. A quasi-periodic attractor may coexist with a stable attractor. Any small changes in the system parameter near the bifurcation point may cause discontinuous variations of mixed convection in confined spaces. In such a case, physiological hazards happen in the ventilated rooms if the system parameters are near the bifurcation point.Dynamical structure and physical mechanism of multiple attractors in mixed convection are investigated using a simplified model in the form of ordinary differential equations. Stability analysis and bifurcation analysis on the dynamical behavior show that when the Archimedes number is fixed in some region, an inverted bifurcation takes place as the Reynolds number is increased. A quasi-periodic attractor may coexist with a stable attractor. Any small changes in the system parameter near the bifurcation point may cause discontinuous variations of mixed convection in confined spaces. In such a case, physiological hazards happen in the ventilated rooms if the system parameters are near the bifurcation point.

Lame function and multi-order exact solutions to nonlinear evolution equations

Abstract In this paper, based on the Lame function and Jacobi elliptic function, the perturbation method is applied to some nonlinear evolution equations, and there many multi-order solutions are derived to these nonlinear evolution equations.

A systematical way to find breather lattice solutions to the positive mKdV equation

In this paper, dependent and independent variable transformations are introduced to solve the positive mKdV equation systematically by using knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the positive mKdV equation, including many kinds of breather lattice solutions.

Power series expansion method and its applications to nonlinear wave equation

The power series expansion method is proposed and applied, just like the reductive perturbation method, to reduce the complicated nonlinear equation or set of equations to be the one that can be found the exact solutions.