Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property
Nobutaka Nakazono
备注：
Some text overlap with our paper arXiv:2102.00684 is caused by our use here of basically the same definition of the CABC property, but the results are totally different
The lattice sine-Gordon equation is an integrable partial difference equation on ${\mathbb Z}^2$, which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property.
A chain of transformations is found which relates one new integrable case of the generalized short pulse equation of Hone, Novikov and Wang [arXiv:1612.02481] with the sine-Gordon equation.
We have studied the solutions of the combined sine-cosine-Gordon Equation found by Wazwaz (App. Math. Comp. 177, 755 (2006)) using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between the phase of the combined sine-cosine-Gordon equation and the parameter in these solutions. Its applications in physical systems are also discussed
It has been unknown whether Hirota's discrete Korteweg-de Vries equation and the lattice sine-Gordon equation have the consistency around a cube (CAC) property. In this paper, we show that they have the CAC property. Moreover, we also show that they can be extended to systems on the 3-dimensional integer lattice.
We examine the question of the integrability of the recently defined $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via appropriate auto-Bäcklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.
We study the local lattice integrable regularization of the Sine-Gordon model written down in terms of the lattice Bose-operators. We show that the local spin Hamiltonian obtained from the six-vertex model with alternating inhomogeneities in fact leads to the Sine-Gordon model in the low-energy limit. We show that the Bethe Ansatz results for this model lead to the correct general relations for different critical exponents of the coupling constant.
We consider the reflectionless transport of sine-Gordon solitons on a line. Transparent boundary conditions for the sine-Gordon equation on a line are derived using the so-called potential approach. Our numerical implementation of these novel boundary conditions proves the absence of the backscattering in transmission of sine-Gordon solitons through the boundary of the considered finite domains.
We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space $H^2$. Our analysis relies on local well-posedness results of Schäfer & Wayne, the correspondence of the short-pulse equation to the sine-Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and global well-posedness of the sine-Gordon equation in an appropriate function space.
It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the ``negative'' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector Nonlinear Schrodinger equations appear as lowest negative and second positive flows within the extended hierarchy. This is fully analogous to the well-known connection between the sine-Gordon and mKdV equations within the extended mKdV hierarchy. A general formalism for a Toda-like symmetry occupying the ``negative'' sector of sl(N) constrained KP hierarchy and g…
Application of our algebraic approach to Liouville integrable defects is proposed for the sine-Gordon model. Integrability of the model is ensured by the underlying classical r-matrix algebra. The first local integrals of motion are identified together with the corresponding Lax pairs. Continuity conditions imposed on the time components of the entailed Lax pairs give rise to the sewing conditions on the defect point consistent with Liouville integrability.
In this paper, from the algebraic reductions from the Lie algebra $gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many multi-component Sine-Gordon type and Sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon equations which can generate new solutions from seed solutions. To see the $Z_n$-systems clearly, we consider the $Z_2$-Sine-Gordon and $Z_3$-Sine-Gordon equations explicitly including their Bäcklund transformations, the nonlin…
We formulate and discuss integrable analogue of the sine-Gordon equation on arbitrary time scales. This unification contains the sine-Gordon equation, discrete sine-Gordon equation and the Hirota equation (doubly discrete sine-Gordon equation) as special cases. We present the Lax pair, check compatibility conditions and construct the Darboux-Bäcklund transformation. Finally, we obtain a soliton solution on arbitrary time scale. The solution is expressed by the so called Cayley exponential function.
We reduce the vectorial binary Darboux transformation for the Manin-Radul supersymmetric KdV system in such a way that it preserves the Manin-Radul-Mathieu supersymmetric KdV equation reduction. Expressions in terms of bosonic Pfaffians are provided for transformed solutions and wave functions. We also consider the implications of these results for the supersymmetric sine-Gordon equation.
Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems i…
Initial-boundary value problems for complex sine-Gordon and sine-Gordon equations in a semi--strip are treated. The evolution of the Weyl function and a uniqueness result are obtained for complex sine-Gordon equation. The evolution of the Weyl function as well as an existence result and a procedure to recover solution are given for sine-Gordon equation. It is shown that for a wide class of examples the solutions of the sine-Gordon equation are unbounded in the quarter-plane.
We treat the lattice sine-Gordon equation and two of its generalised symmetries as a compatible system. Elimination of shifts from the two symmetries of the lattice sine-Gordon equation yields an integrable NLS-type system. An auto-Bäcklund transformation and a superposition formula for the NLS-type system is obtained by elimination of shifts from the lattice sine-Gordon equation and its down-shifted version. We use the obtained formulae to calculate a superposition of two and three elementary solutions.
In this paper we study the sine-Gordon and the Liouville hierarchies in laboratory coordinates from a bi-Hamiltonian point of view. Besides the well-known local structure these hierarchies possess a second compatible non-local Poisson structure.
The permutability of two Backlund transformations is employed to construct a non linear superposition formula and to generate a class of solutions for the N=2 super sine-Gordon model.
A new way for finding analytical solutions of the three-dimensional sine-Gordon equation is presented. The method is based on the established relation between the solutions of the three-dimensional wave equation and solutions of the three-dimensional sine-Gordon equation. Some examples of the solutions thus obtained are show.
In an early work, Bernoulli shift dynamics of submanifolds was established in a neighborhood of a homoclinic tube. In this article, we will present a concrete example: sine-Gordon equation under a quasi-periodic perturbation.
It is well known that the classical string on a two-sphere is more or less equivalent to the sine-Gordon model. We consider the nonabelian dual of the classical string on a two-sphere. We show that there is a projection map from the phase space of this model to the phase space of the sine-Gordon model. The corresponding Poisson structure of the sine-Gordon model is nonlocal with one integration.
The aim of this paper is to introduce the Inverse Scattering Method for later studies of some problems in nonlinear dynamics, and describe the kink solution of the Sine Gordon Equation using the Inverse Scattering Method as a methodological example, the soliton solution is well known.
This paper considers the whole hierarchy of bi-Hamiltonian integrable equations associated to each of the Short-Pulse (SP) equation and the Sine-Gordon (SG) equation. We prove that the transformation that relates the SP equation with the SG equation also serves to establish the correspondence between their flows and Hamiltonian conservation laws in respective hierarchy.
We study a discrete Darboux transformation and construct the multi-soliton solutions in terms of ratio of determinants for integrable discrete sine-Gordon equation. We also calculate explicit expressions of single, double, triple, quad soliton solutions as well as single and double breather solutions of discrete sine-Gordon equation. Dynamical features of discrete kinks and breathers have also been illustrated.
By studying the {\it internal} Riemannian geometry of the surfaces of constant negative scalar curvature, we obtain a natural map between the Liouville, and the sine-Gordon equations. First, considering isometric immersions into the Lobachevskian plane, we obtain an uniform expression for the general (locally defined) solution of both the equations. Second, we prove that there is a Lie-Bäcklund transformation interpolating between Liouville and sine-Gordon. Third, we use isometric immersions into the Lobachevskian plane to describe sine-Gordon N-solitons explicitly.
It was recently noted how the classical sine-Gordon theory can support discontinuities, or `defects', and yet maintain integrability by preserving sufficiently many conservation laws. Since soliton number is not preserved by a defect, a possible application to the construction of logical gates is suggested.
We analyze a generalization of the sine-Gordon equation in laboratory coordinates on the half-line. Using the Fokas transform method for the analysis of initial-boundary value problems for integrable PDEs, we show that the solution $u(x,t)$ can be constructed from the initial and boundary values via the solution of a $2\times 2$-matrix Riemann-Hilbert problem.
The method of Darboux transformation, which is applied on cnoidal wave solutions of the sine-Gordon equation, gives solitons moving on a cnoidal wave background. Interesting characteristics of the solution, i.e., the velocity of solitons and the shift of crests of cnoidal waves along a soliton, are calculated. Solutions are classified into three types (Type-1A, Type-1B, Type-2) according to their apparent distinct properties.
In this paper, local and nonlocal complex reduction of a discrete and a semi-discrete negative order Ablowitz-Kaup-Newell- Segur equations is studied. Cauchy matrix type solutions, including soliton solutions and Jordan-block solutions, for the resulting local and nonlocal complex discrete and semi-discrete sine-Gordon equations are constructed. Dynamics of 1-soliton solution are analyzed and illustrated.
Recently it was discussed the Inverse Scattering Method, Part I. (paper I.) . It was a methodological Part with an example - soliton (kink) solution of the Sine-Gordon Equation. The aim of the paper I. was to introduce the Inverse Scattering Method for later studies of some problems in nonlinear dynamics. As a methodological example we described how to solve the Sine-Gordon Equation using the Inverse Scattering Method to obtain a soliton. This soliton solution is well known. In this paper we discuss another methodological example: the solution of the Sine-Gordon Equation using the Inverse Scat…
We implement the numerical inverse scattering transform (NIST) for the sine-Gordon equation in laboratory coordinates on the real line using the method developed by Trogdon, Olver and Deconinck. The NIST allows one to compute the solution at any x and t without having spatial discretization or time-stepping. The numerical implementation is fully spectrally accurate. With the help of the method of nonlinear steepest descent, the NIST is demonstrated to be uniformly accurate.
Recently, the method of one-dimensional maps was introduced as a means of generating exceptional discretisations of the $ϕ^4$-theories, i.e., discrete $ϕ^4$-models which support kinks centred at a continuous range of positions relative to the lattice. In this paper, we employ this method to obtain exceptional discretisations of the sine-Gordon equation (i.e. exceptional Frenkel-Kontorova chains). We also use one-dimensional maps to construct a discrete sine-Gordon equation supporting kinks moving with arbitrary velocities without emitting radiation.
Stochastic processes associated with traveling wave solutions of the sine-Gordon equation are presented. The structure of the forward Kolmogorov equation as a conservation law is essential in the construction and so is the traveling wave structure. The derived stochastic processes are analyzed numerically. An interpretation of the behaviors of the stochastic processes is given in terms of the equation of motion.
We construct explicit multivortex solutions for the complex sine-Gordon equation (the Lund-Regge model) in two Euclidean dimensions. Unlike the previously found (coaxial) multivortices, the new solutions comprise $n$ single vortices placed at arbitrary positions (but confined within a finite part of the plane.) All multivortices, including the single vortex, have an infinite number of parameters. We also show that, in contrast to the coaxial complex sine-Gordon multivortices, the axially-symmetric solutions of the Ginzburg-Landau model (the stationary Gross-Pitaevskii equation) {\it do not} be…
It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency.
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 Evans-Krein function theory 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 …
Surfaces of constant negative curvature in Euclidean space can be described by either the sine-Gordon equation for the angle between asymptotic directions, or a Monge-Ampere equation for the graph of the surface. We present the explicit form of the correspondence between these two integrable non-linear partial differential equations using their well-known properties in differential geometry. We find that the cotangent of the angle between asymptotic directions is directly related to the mean curvature of the surface. This is a Backlund-type transformation between the sine-Gordon and Monge-Ampe…
We construct explicit multivortex solutions for the first and second complex sine-Gordon equations. The constructed solutions are expressible in terms of the modified Bessel and rational functions, respectively. The vorticity-raising and lowering Backlund transformations are interpreted as the Schlesinger transformations of the fifth Painleve equation.
The matrix sine-Gordon theory, a matrix generalization of the well-known sine-Gordon theory, is studied. In particular, the $A_{3}$-generalization where fields take value in $SU(2)$ describes integrable deformations of conformal field theory corresponding to the coset $SU(2) \times SU(2) /SU(2)$. Various classical aspects of the matrix sine-Gordon theory are addressed. We find exact solutions, solitons and breathers which generalize those of the sine-Gordon theory with internal degrees of freedom, by applying the Zakharov-Shabat dressing method and explain their physical properties. Infinite c…
We prove that the tau-function of the integrable discrete sine-Gordon model apart from the "standard" bilinar identities obeys a number of "non-standard" ones. They can be combined into a bivector 3-dimensional difference equation which is shown to contain Hirota's difference analogue of the sine-Gordon equation and both auxiliary linear problems for it. We observe that this equation is most naturally written in terms of the quantum R-matrix for the XXZ spin chain and looks then like a relation of the "vertex-face correspondence" type.
The sine-Gordon equation, used as the representative nonlinear wave equation, presents a bistable behavior resulting from nonlinearity and generating hysteresis properties. We show that the process can be understood in a comprehensive analytical formulation and that it is a generic property of nonlinear systems possessing a natural band gap. The approach allows to discover that sine-Gordon can work as an it ideal switch by reaching a transmissive regime with vanishing driving amplitude.
We consider discrete sine-Gordon equation on branched domains. The latter is modeled in terms of the metric graphs with discrete bonds having the form of the branched 1D chains. Exact analytical solutions of the problem are obtained for special case of the constraints given by in terms of simple sum rule. Numerical solution is obtained when the constraint is not fulfilled.
It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency.
In the talk, the relationship between black holes in Jackiw-Teitelboim(JT) dilaton gravity and solitons in sine-Gordon field theory is described. The well-known connection between solutions of the sine-Gordon equation and constant curvature metrics is reviewed and expanded. In particular, it is shown that solutions to the dilaton field equations for a given metric in JT theory also solve the sine-Gordon equation linearized about the corresponding soliton. Next, it is shown that from the B${\ddot a}$cklund transformations relating different soliton solutions, it is possible to construct a flat …
