On the sine-Gordon equation with a self-consistent source

We use the method of the inverse scattering problem to solve the sine-Gordon equation with a self-consistent source which corresponds to moving eigenvalues of the corresponding spectral problem.     

Numerical solution of the sine-Gordon equation

Two new difference schemes are proposed for the sine-Gordon equation. They have the advantage that there is a discrete energy which is conserved. Their stability and convergence is proved. Also we show some numerical computations using one of these schemes.     

Solutions of the three-dimensional sine-Gordon equation

We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009.     

完整Coriolis力作用下带有外源强迫的非线性ZK方程

在正压流体中,从包含完整Coriolis参数的准地转位涡方程出发,在弱非线性长波近似下,采用多时空尺度和摄动方法,推导出大气非线性Rossby波振幅演变满足带有外源强迫的二维Zakharov-Kuznetsov(ZK)方程.然后利用Jacobi椭圆函数展开法,求解了ZK方程的椭圆正弦波解和孤立波解.分析结果表明,Coriolis参数的水平分量将影响二维Rossby波传播的频率特征,而外源不仅对二维Rossby波动的传播的频率有影响,对振幅也产生一个调制作用.     

On singularity of a nonlinear variational sine-Gordon equation

We study a sine-Gordon-type of nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude. This equation arises naturally from long waves on a dipole chain in the continuum limit, which provides a crude model for some polymers. Using characteristic methods, we describe a blow-up result for the one-dimensional nonlinear variational sine-Gordon equation, which shows that smooth solutions breakdown in finite time.     

Kink asymptotics of the perturbed sine-Gordon equation

The leading term and the first correction to the solution of the Goursat problem for the perturbed sine-Gordon equation are constructed at times of the order of the inverse of the perturbation parameter. The equation for the modulation of the first correction in the continuous spectrum is obtained.Institute of Mathematics with the Computational Center of the Bashkir Scientific Center of the Urals Branch of the Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 1, pp. 39–48, October, 1992.     

Perturbed soliton solutions of the sine-Gordon equation

Soliton solutions of the sine-Gordon classical equation are numerically studied. It is shown that considerable perturbations in these solutions lead to the formation of new solution forms that exhibit soliton properties in interactions. The study is performed for kinks and breathers obtained by solving problems with suitable initial data. The underlying numerical technique combines the fourth-order Runge-Kutta method with the quasi-spectral Fourier method.     

Some solutions of discrete sine-Gordon equation

In this paper, a series of exact solutions of discrete sine-Gordon equation are obtained by the different transformations and symbolic computation.     

Attractors for a plate equation with nonlocal nonlinearities

This paper contains results on well‐posedness, stability, and long‐time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of R ⁿ with smooth boundary, γ ,ρ ?0 and are nonlocal functions. The main result states that the dynamical system {S (t )}_{t ?0} associated with this problem has a compact global attractor. In addition, in the limit case γ  = 0, it is also shown that {S (t )}_{t ?0} has a finite dimensional global attractor by using an approach on quasi‐stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.     

Attractors for a von Karman equation with memory

In this paper a von Karman equation with memory,utt + α?2u- γ?utt- integral from n=-∞ to t μ(t- s)?2u(s)ds = [u, F(u)] + h is considered. This equation was considered by several authors. Existing results are mainly devoted to global existence and energy decay. However, the existence of attractors has not yet been considered. Thus, we prove the existence and uniqueness of solutions by using Galerkin method, and then show the existence of a finitedimensional global attractor.     

Bound state solutions of the elliptic sine-Gordon equation

We present a detailed investigation of finite-energy solutions with point-like singularities of the elliptic sine-Gordon equation in a plane. Such solutions are of the bound-state type in the sense of scalar field theory. If the solution has a unique singularity, then it behaves as a soliton-like annular wave packet at a large distance from the singularity. The effective radius of this wave packet is evaluated both analytically and numerically for axially symmetric solutions. The analytical investigation is based on the method of isomonodromy deformations for the third Painlevé equation, which singles out these solutions as separatrices of the manifold of general solutions (with infinite energy). Exact analytical estimates provide a tool for investigating bound-state solutions of the nonintegrable sine-Gordon equation with a nonzero right-hand side. More precisely, for large-intensity fields at the singularity, we derive the critical forcing that allows the existence and stability of a bound state. As an illustration, we consider two applications: a large-area Josephson junction and a nematic liquid crystal in a rotating magnetic field. For each of the examples, we evaluate the critical values of the field that allow finite-energy regimes. These are in good agreement with numerical and experimental data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 15–31, April, 1997.     

Recursion operator for the intrinsic generalized sine-Gordon equation

We find the inverse and direct recursion operator for the intrinsic generalized sine-Gordon equation in any number n > 2 of independent variables. Among the flows generated by the direct operator we identify a higher-dimensional analogue of the pmKdV equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 117–128, 2006.