Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg?CMarchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schr?dinger equation are also derived.     

An evolution equation in phase space and the Weyl correspondence

The Weyl correspondence that associates a quantum-mechanical operator to a Hamiltonian function on phase space is defined for all tempered distributions on R². The resulting Weyl operators are shown to include most Schroedinger operators for a system with one degree of freedom. For each tempered distribution, an evolution equation in phase space is defined that is formally equivalent to the dynamics of the Heisenberg picture. The evolution equation is studied both through a separation of variables technique that expresses the evolution operator as the difference of two Weyl operators and through the geometric properties of the distribution. For real tempered distributions with compact support the evolution equation has a unique solution if and only if the Weyl equation does. The evolution operator has skew-adjoint extensions that solve the evolution equation if the distribution satisfies an orthogonal symmetry condition.     

Construction of the Solution of the Inverse Spectral Problem for a System Depending Rationally on the Spectral Parameter,Borg–Marchenko-Type Theorem and Sine-Gordon Equation

Weyl theory for a non-classical system depending rationally on the spectral parameter is treated. Borg–Marchenko-type uniqueness theorem is proved. The solution of the inverse problem is constructed. An application to sine-Gordon equation in laboratory coordinates is given.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

An Integrable Discretization of a 2+1-dimensional Sine-Gordon Equation

We show that the complex discrete BKP equation that has been recently identified as an integrable discretization of the 2+1-dimensional sine-Gordon system introduced by Konopelchenko and Rogers admits a natural reduction to a discrete 2+1-dimensional sine-Gordon equation. We discuss three important properties of this equation. First, it may be interpreted as a superposition principle associated with a constrained Moutard transformation. Second, the complexified discrete sine-Gordon equation constitutes an eigenfunction equation for the discrete sine-Gordon system. Third, we derive a form of the equation in terms of trigonometric functions that has been studied by Konopelchenko and Schief in a discrete geometric context. A discrete Moutard transformation for the discrete sine-Gordon equation and the corresponding Bäcklund equations are also recorded.     

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Rectangular matrix solutions of the defocusing nonlinear Schrödinger equation (dNLS) are studied in quarter-plane and semi-strip. Evolution of the corresponding Weyl–Titchmarsh (Weyl) function is described in terms of the initial Weyl function and boundary conditions. In the next step, the initial Weyl function is recovered (for the quarter-plane case) from the long-time asymptotics of the wave function considered at the boundary. Thus, it is shown that the evolution of the Weyl function is uniquely defined by the boundary conditions. Moreover, a procedure to recover solutions of dNLS (uniquely defined by the boundary conditions) is given. In a somewhat different way, the same boundary value problem is also dealt with in a semi-strip (for the case of a quasi-analytic initial condition).     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction

The focus of study is the nonlinear discrete sine-Gordon equation, where the nonlinearity refers to a nonlinear interaction of neighbouring atoms. The existence of travelling heteroclinic, homoclinic and periodic waves is shown. The asymptotic states are chosen such that the action functional is finite. The proofs employ variational methods, in particular a suitable concentration-compactness lemma combined with direct minimisation and mountain pass arguments.     

Sine-Gordon方程的多辛Leap-frog格式

非线性发展方程由于具有多种形式的解析解而吸引着众多的研究者,借助多辛保结构理论研究了Sine-Gordon方程的多辛算法．利用Hamilton变分原理,构造出了sine-Gordon方程的多辛格式;采用显辛离散方法得到了Leap-frog多辛离散格式,该格式满足多辛守恒律;数值结果表明leap-frog多辛离散格式能够精确地模拟sine-Gordon方程的孤子解和周期解,模拟结果证实了该离散格式具有良好的数值稳定性．     

Approximate controllability of evolution systems with nonlocal conditions

We study the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to show the application of our result.     

One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation

This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. It is shown that for any positive damping and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random attractor is a one-dimensional random horizontal curve regardless of the strength of noise. Hence its dynamics is not chaotic. It is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large, which implies that the solutions tend to oscillate with the same frequency eventually and the so-called frequency locking is successful.     

A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation

We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill’s equation. Applying the Evans-Krein function theory of Kollár and Miller (SIAM Rev 56(1):73–123, 2014) to our Evans function, we provide a new method for computing the Krein signatures of simple characteristic values of the linearised sine-Gordon equation. By varying the Floquet exponent parametrising the quasi-periodic solutions, we compute the linearised spectra of periodic travelling wave solutions of the sine-Gordon equation and track dynamical Hamiltonian–Hopf bifurcations via the Krein signature. Finally, we show that our new Evans function can be readily applied to the general case of the nonlinear Klein–Gordon equation with a non-periodic potential.

     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

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.     

Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm

Recently, we have presented a sine-Gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations. In this paper we further develop the method to be the discrete sine-Gordon expansion method in nonlinear differential-difference equations, in particular, discrete soliton equations. We choose the modified Volterra lattice and Volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained. Moreover some figures display the profiles of the obtained solutions. Our method can be also applied to other discrete soliton equations.     

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

Integration of the sine-Gordon equation with a self-consistent source of the integral type in the case of multiple eigenvalues

The paper is dedicated to the evolution of the scattering data for a Dirac-type nonself-adjoint operator with multiple eigenvalues whose potential is a solution of the sine-Gordon equation with a self-consistent source of the integral type.     

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.     

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer–Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations.     

General existence principles for Stieltjes differential equations with applications to mathematical biology

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.