Mixed solutions of the sine-Gordon equation

The aim of this note is to present an explicit formula for the mixed solution of the sine-Gordon equation, i.e. the solution describing a multisoliton process on a background of multiphase quasi-periodic process.     

The solutions of the two-dimensional sine-Gordon equation

Three-parameter families of the solutions of sine-Gordon equation in two dimensions are presented.     

Solitions and separable elliptic solutions of the sine-Gordon equation

We point out that the two-soliton (antisoliton) solutions of the sine-Gordon equation may be obtained as limiting cases of a separable, two-parameter family of elliptic solutions. The solitons are found on the boundary of the parameter space for the elliptic solutions when the latter are considered over their usual complex domain.     

Exact and numerical solutions to the perturbed sine-Gordon equation

We present numerical and analytic solutions to the perturbed sine-Gordon equation, which models long Josephson tunnel junctions. We make comparisons between numerical results and results obtained from perturbational methods. We present unstable, analytic kink solutions to the equation and further a solution, which is an array of kinks, corresponding to a solution, where the current through the junction is larger than the critical current.     

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

We construct the Hirota bilinear form of the nonlocal Boussinesq(nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nl Bq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nl Bq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form.The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.     

Approximate treatment of two soliton solutions of the sine-Gordon equation

The so called breather solution of the sine-Gordon equation is phenomenologically described by an appropriately choosen potential acting between two particles. For some applications the method proves to be equivalent to other classical and quantum calculations.     

An approach to radially symmetrical solutions of the sine-Gordon equation

The solution φ(r, t) of the radially symmetric sine-Gordon equation is considered in three and two spatial dimensions for initial curves, analogous to a 2π-kink, in the expanding and in the shrinking phase, for R(t)j? R(0). It is shown that the parameterization φ(r, t) = 4 arcian exp[γ(r?R(0)] + x(r, t), where R(t) describes the exact propagation of the maximum of φ,(r, t), is suitable. Using an appoximate differential equation, recently given for the propagation of the solitary ring wave, a rough analytic approximation for the correction function x(r = R(t), t) is found and tested numerically. A relationship between the fluctuations in x(r = R(t), t) and those in $\text{R}\text{?}\text{(t), t)}\text{and}\text{R(t)}$ explains why the solitary wave is almost stable. From x(r = R(t), t) and the supposition x(1, t) ≈ x(∞, t) ≈ 0 an assymetry in φ_r(r, t) with respect to r = R(t) is predicted. It also exhibits fluctuations corresponding to those in x(r = R(t), t). The condition for validity of this approximation apparently is also a limit for the stability of the solitary ring wave.     

Quasi-periodic solutions of the integrable dynamical systems related to Hill's equation

We present finite-gap solutions to the Garnier system and to the g-dimensional anisotropic harmonic oscillator in a radial quartic potential. The relationship between these solutions and solutions of Neumann-type dynamical systems is discussed.     

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε>0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε. This sequence of exact solutions is analogous to that of the well-known N-soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε↓0 one observes the appearance of nonlinear caustics, i.e. curves in space-time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases.In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L¹-Sobolev initial data, and Appendix B establishes the well-posedness for L^p-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).     

A generalized exp-function method for multiwave solutions of sine-Gordon equation

In this paper, the exp-function method is generalized to sine-Gordon (sG) equation and single-, double- and three-wave solutions are obtained. It is shown that the generalized exp-function method combined with appropriate anstaz may provide a straightforward, effective and alternative method for constructing multiwave solutions of some nonlinear partial differential equations.     

Interaction between solitary wave solutions to a double sine-Gordon equation

Solitary wave solutions to a double sine-Gordon equation are examined by means of numerical computations. Quasi-solitons as well as solitary wave solutions are found. The break up of initial shapes is examined.     

Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation

Expanding rotationally symmetric solitary wave solutions to the three- and two-dimensional sine-Gordon equation (ring waves) are shown to reach a maximum extension and then shrink. The phenomenon, which we denote a return effect, is investigated numerically and analytically.     

Separable solutions of the sine-Gordon equation in terms of a Painlevé transcendent

We show that, when expressed in characteristic coordinates, the two-dimensional sine-Gordon equation may be reduced to a separable form via the use of a fifth Painlevé transcendent.     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

The dissipative sine-Gordon equation

Application of the decomposition method to the generalizations of the sine-Gordon equation provide efficient analytic solution without linearization or perturbation.     

Bethe-Salpeter equation with the sine-Gordon interaction

An attempt is made to study the interaction Hamiltonian,H _int =Gψ ²(x)U(φ(x)) in the Bethe-Salpeter framework for the confined states of theψ particles interactingvia the exchange of theU field, whereU(φ) = cos (gφ). An approximate solution of the eigenvalue problem is obtained in the instantaneous approximation by projecting the Wick-rotated Bethe-Salpeter equation onto the surface of a four-dimensional sphere and employing Hecke’s theorem in the weak-binding limit. We find that the spectrum of energies for the confined states,E =2m+B (B is the binding energy), is characterized byE ∼n ⁶, wheren is the principal quantum number.     

Painleve analysis of the perturbed sine-Gordon equation

We are reporting on numerical investigations of a seven-variable model corresponding to a class of chemical reactions which exhibit, as a function of the control parameter, a sequence of periodic and chaotic states strikingly similar to that observed in bench experiments. This scenario involves period-doubling cascades, tangent bifurcations and intermittency, in good agreement with a dynamical evolution predicted by a multi-humped one-dimensional map. This strongly suggests an interpretation of the strange-attractor-like behavior observed along such paths, in terms of the chaotic behavior which occurs nearby homoclinic conditions.     

（2＋1）维sine—Gordon方程的三种函数混合解

基于一构造的Wronskian形式展开法,获得了（2＋1）维sine—Gordon方程的一系列新形式的混合解．这些解是三角函数、双曲函数及Jacobi椭圆函数等三种函数的组合形式．     

(n+1)维双Sine-Gordon方程的新精确解

给出包含第一种椭圆方程的三角函数型辅助方程及其解的叠加公式.在一般函数变换下,借助符号计算系统Mathematica,构造了(n+1)维双sine-Gordon方程新的Jacobi椭圆函数精确解.这些解包括了行波变换下的Jacobi椭圆函数精确解、精确孤立波解和三角函数解.