Perturbation of solitons due to power law nonlinearity

The generalization of solitons to a non-Kerr law media has been studied in this paper along with its perturbation. In particular, the higher nonlinear Schrödinger's equation (NLSE) due to power law nonlinearity is considered. The method of multiple-scales is used to study this equation in presence of a perturbation term. We show that, by newly introducing a proper definition of the phase of the soliton, for the first time, one can obtain the corrections to the pulse where the usual soliton perturbation approach fails.     

Mathematical structure of topological solitons due to the Sine-Gordon Equation

This paper studies the mathematical structure of the kink soliton solution to the generalized dispersive SGE. All derivatives are found and the trigonometric functions of the kink are found for a generalized height.     

How to excite the internal modes of sine-Gordon solitons

We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton behaves as a deformable (non-rigid) object. A soliton moving in an array of inhomogeneities can also present sustained oscillations of its width. There are very important phenomena (like the soliton–antisoliton collisions) where the existence of internal modes plays a crucial role. We show that, under some conditions, the dynamics of the soliton shape modes can be chaotic. A short report of some of our results has been published in [Phys. Rev. E 65 (2002) 065601(R)].     

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article: F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

New approach to the solution of the classical sine-Gordon equation and its generalizations

We obtain new exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation. The three-dimensional solutions depend on an arbitrary function F(α) whose argument is a function α(x, y, z, t). The ansatz α is found from an equation linear in (x, y, z, t) whose coefficients are arbitrary functions of α that should satisfy a system of algebraic equations. By this method, we solve the classical and a generalized sine-Gordon equation; the latter additionally contains first derivatives with respect to (x, y, z, t). We separately consider an equation that contains only the first derivative with respect to time. We present approaches to the solution of the sine-Gordon equation with variable amplitude. The considered methods for solving the sine-Gordon equation admit a natural generalization to the case of integration of the same types of equations in a space of arbitrarily many dimensions.     

Weingarten surfaces and sine-Gordon equation

A relation between Weingarten surfaces with conditionK - 2mH +m ² +l ² = 0 and solutions of sine-Gordon equations is established.     

A geometric interpretation of stationary-wave type solutions of the sine-Gordon equation

Using the connection of solutions of the sine-Gordon equationz _xy =sinz with the construction of a grid of asymptotic lines on surfaces of constant negative curvature inE ³ we show that those surfaces and only those that have curvature –1 and any two asymptotic lines of a single family of which are congruent correspond to solutions of stationary-wave typez=z(x+by).Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 81–87.     

Particular solutions to equations of sine-Gordon type

We present a direct method for finding particular solutions of equations of sine-Gordon type, including single sinh-Gordon equation, single sine-Gordon equation, double sinh-Gordon and double sine-Gordon equations.     

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.     

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.     

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.     

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.     

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.     

Tension spline solution of nonlinear sine-Gordon equation

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders O(k²+k²h²+h²)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2}) and O(k²+k²h²+h⁴)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4}). At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.