Nonlinear sine-Gordon Equation Research Articles

A New Approach to Solve Nonlinear Wave Equations**The project supported by State Key Project of National Natural Science Foundation of China (No. 40035010) and National Natural Science Foundation of China (No. 40175016)

From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave 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 conditions.

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.

Conformally Flat Hypersurfaces of E 4

We consider 3-dimensional conformally flat hypersurfaces of E4 with 2 different principal curvatures such that the coordinate directions are principal directions. We describe explicitly those which allow an immersion with constant mean curvature. They are shown to be in close correspondence with solutions of the nonlinear integrable sine-Gordon and sinh-Gordon equations. Conversely, this provides a geometrical characterization for this particular class of conformally flat hypersurfaces of E4.

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.

Complex Fermi-Pasta-Ulam Recurrence and the Problem of Information Storage in a Neuron

A qualitative model of a neuron was studied theoretically and experimentally. A coupled system of nonlinear Schrodinger (NLS) and sine-Gordon equations was used for mathematical description of the complex Fermi-Pasta-Ulam recurrence interpreted as a principal information carrier in the model. Computer and analog experiments were devised to verify the proposed concept.

Stochastic propagation of fluxons on Josephson lines

In this paper, the fluxons propagation on Josephson junction is modeled by the nonlinear Sine-Gordon equation which has received much attention in recent years. The stochastic surface current is studied under the effect of stochastic propagation velocity. The probability density function of the surface current is derived using successive random process transformation. Furthermore, the stochastic moments of the surface current are computed for some case studies.

Nonlinear effects in liquid crystals

The effects of nonlinearity in nematic liquid crystals subject to electric and magnetic fields are emphasized. For the Fr\'eedericks transition, the distortion profile is calculated exactly. For the domain modes, the torque equation can be approximated to a high degree by the nonlinear sine-Gordon equation. An exact two-dimensional periodic solution is found whose consequences are in good quantitative agreement with all experimental observations. The possibility of a first-order transition to a vortex pattern is predicted.

Nonlinear stability problems for the sine-Gordon equation

Two stability problems for the nonlinear sine-Gordon equation are studied. The stability of a class of time independent (static) solutions is studied using linear dynamic stability theory. An asymptotic approximation of the nonlinear transient response to small disturbances of an unstable static state is obtained by the two time method. Interpretations of the results are given for the continuous pendulum problem and for the Josephson tunnel junction. A proof of the validity of the asymptotic approximation is given.