Conservation laws and symmetries of generalized sine-Gordon equations

We study some systems of non-linear PDE's (Eqs. 1.1 below) which can be regarded either as generalizations of the sine-Gordon equation or as two-dimensional versions of the Toda lattice equations. We show that these systems have an infinite number of non-trivial conservation laws and an infinite number of symmetries. The second result is deduced from the first by a variant of the Hamiltonian formalism for evolution equations. We also consider some specializations of the systems.     

Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

A mechanical analog for the double sine-Gordon equation

A mechanical analog for the double sine-Gordon equation is proposed and used to analyze solitary solutions for arbitrary parameter values. The extension of this analog to other equations of the multiple sine-Gordon class is also considered.     

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.     

Quasi-periodic solutions of the sine-Gordon equation

Two-phase quasi-periodic solutions of the sine-Gordon equation are discussed. It is shown that the solutions expressed by the multidimensional θ-function form a broader class than the well-known factorised solutions 4 arctan [f(u)g(v)]. Expressions describing the spatial and temporal period are also given.     

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r s^T we introduce a scalar function S^{(i, j)} = s^T K^j (I + M)^?1Kⁱr which is defined as same as in discrete case. S^{(i, j)} satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S^{(i, j)} defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S^{(i, j)} = S^(j,i). The solution M provides t function by t = ∣I + M∣. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.     

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.     

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.     

Lax pairs for certain generalizations of the sine-Gordon equation

We derive the one-parameter family of isospectral linear eigenvalue problems which is the basic tool for treating certain generalized sine-Gordon equations by the inverse scattering method.     

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.     

The solutions of the two-dimensional sine-Gordon equation

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

Flow equation approach to the sine-Gordon model

A continuous sequence of infinitesimal unitary transformations is used to diagonalize the quantum sine-Gordon model for β²∈(2π,∞). This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion: a small expansion parameter is identified and this parameter remains small throughout the entire flow unlike the diverging running coupling constant of perturbative scaling. Our approximation consists in neglecting higher orders in this small parameter. We find very accurate results for the single-particle/hole spectrum in the strong-coupling phase and can describe the full crossover from weak to strong-coupling. The integrable structure of the sine-Gordon model is not used in our approach. Our new method should be of interest for the investigation of nonintegrable perturbations and for other strong-coupling problems.     

Some exact solutions to the sine-Gordon equations

We present some new exact (solitary and traveling) solutions to the sine-Gordon equationsc ²u_xx–u_tt = sinu andcu _xt+ u_tt = sinu, and the dissipative sine-Gordon equationc ²u_xx–u_tt–u_t=₁ sinu+ ₂ sin(2u). Here,, ₁, and ₂ are the coupling constants, real numbers;c is theintrinsic speed of the propagating wave determined by the system; and is the damping factor. The physical implication is briefly discussed.     

Noether's theorem and Steudel's conserved currents for the sine-Gordon equation

A version of Noether's theorem appropriate for the extended Hamilton-Cartan formalism for regular first-order Lagrangians is proposed. Steudel's derivation of an infinite collection of conserved currents for the sine-Gordon equation is presented in this context and it is demonstrated that, as a consequence of the commutativity of the sine-Gordon Bäcklund transformations, the conserved charges corresponding to these currents are in involution with respect to the natural Poisson bracket provided by the formalism. Thus one obtains the formal complete integrability of the sine-Gordon equation as a consequence of the properties of the Bäcklund transformation.     

A physics-constrained deep residual network for solving the sine-Gordon equation

Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues.First,it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well.Second,the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks.In this paper,we propose a new architecture that combines the deep residual neural network with some underlying physical laws.Using the sine-Gordon equation as an example,we show that the numerical result is in good agreement with the exact soliton solution.In addition,a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.     

Bosonic symmetries of the Dirac equation

We have found on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass the new physically meaningful features of this equation. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found, among which (together with the 32-dimensional pure matrix algebra of invariance) the new spin s=(1,0) multiplet Poincaré symmetry is proved. In order to carry out the corresponding proofs a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.