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In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)\ast}$-type discrete Painlev\'e equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra., Comment: arXiv admin note: text overlap with arXiv:1906.06650

In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.

We present new oscillation tests for the PΔE a u ( m + 1 , n ) + b u ( m , n + 1 ) − c u ( m , n ) + p ( m , n ) u ( m − τ , n − σ ) = 0 , m , n = 0 , 1 , ⋯ , where a, b, c ∈ (0, ∞) and τ, σ ∈ {0, 1, ⋅⋅⋅}, and {p(m, n)} ⊂ [0, ∞). Our main result improves some of the well-known results in the literature. We also present a numerical example, where all the previous results in the literature fail to deliver an answer.

In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show a few examples, both in partial differential and partial difference equations where this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries.

This paper deals with the relativistic, quantized electromagnetic and Dirac field equations in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. In the consequent relativistic quantum electrodynamics, the corresponding Feynman diagrams and S#-matrix elements are derived. In the special case of electron-electron scattering (Moller scattering), the explicit second order element is deduced. Moreover, assuming the slow motions for two external electrons, the approximation of yields a divergence-free Coulomb potential., Comment: 19 Pages, 2 tables, 3 figures

We derive the determining equations for the N th-order generalized symmetries of partial difference equations defined on d consecutive quadrilaterals on the lattice using the theory of integrability conditions. We provide their algebraic formulation and develop the necessary theoretical framework for their analysis along with a systematic method for solving functional equations of the form T ( f ) + A f + B = 0 . Our approach is algorithmic and can be easily implemented in symbolic computations. We demonstrate our approach by deriving the symmetries of various equations and discuss certain applications and extensions of the theory.

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.

We discuss the non autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its B\"acklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the $H^4$ trapezoidal and the $H^6$ families are linearizable and in a few examples we show how we can effectively linearize them.

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation., 17 pages. 1 figure

In this paper, we develop criteria for oscillation of the partial difference equation with continuous variables of the form Au(x+a,y+b) + Bu(x+a,y) + Cu(x,y+b) - u(x,y) + D(x,y)u(x-τ,y) + E(x,y)u(x,y-σ) + F(x,y)u(x-τ,y-σ) = 0, where D,E,F ∈ C(R+ × R+, R+ - {0}), A,B and C are nonnegative constants, a, b, τ and σ are positive constants. Several examples which dwell upon the importance of our results are also included.

Motivated by the classical one-dimensional diffusion equation, u t ku xx = f(x,t), xϵR, t≥0, u(x,0)=ϵ(x), xϵR, we find the unique solution of the partial difference equation δ t u(x,t)-kδ 2 x,x u(x-1,t)=f(x,t), xϵZ, t≥0, u(x,0)=ϵ(x), xϵZ, where k ϵ (0, 1). Our approach is to construct the Green's function for the appropriate partial difference operator, and almost immediately we establish an existence and uniqueness result for the discrete nonhomogenous problem.

This paper introduces a technique for obtaining the conservation laws of a given scalar partial difference equation with two independent variables. Unlike methods that are based on Nother's theorem, this new technique does not use symmetries. Neither does it require the difference equation to have any special structure, such as a Lagrangian, Hamiltonian or multisymplectic formulation. Instead, it uses a discrete analogue of the variational complex.

We offer sufficient conditions for the oscillation of all solutions of the partial difference equations y(m - 1,n) + β(m,n)y(m, n - 1) -δ(m,n)+ P(m,n,y(m + k,n + l)) = Q(m,n,y(m + k,n + l)) and (y(m - 1,n)+ β(m,n)y(m,n - 1) - δ(m,n)y(m,n) + $$\mathop \Sigma \limits_{i = 1}^\tau $$ Pi(m,ny(m + ki,n + li)) = $$\mathop \Sigma \limits_{i = 1}^\tau $$ Qi(m,n,y9m + ki,n + li)). Several examples which dwell upon the importance of our results are also included.

A Riesz difference is defined by the use of the Riemann-Liouville differences on time scales. Then the definition is considered for discrete fractional modelling. A lattice fractional equation method is proposed among which the space variable is defined on discrete domains. Finite memory effects are introduced into the lattice system and the numerical formulae are given. Adomian decomposition method is adopted to solve the fractional partial difference equations numerically.

In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples.

Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products., 24 pages

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A(1), A(2) and A(3) linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A(3) C-integrability conditions can be linearized by a Mobius transformation.

In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations "of KdV type" that were known since the late 1970s and early 1980s. In this paper we review the construction of soliton solutions for the KdV type lattice equations and use those results to construct N-soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment., Comment: 40 pages, 3 figures, submitted to Journal of Physics A: Mathematical and Theoretical, Special issue on nonlinearity and geometry: connections with integrability

The lattice Gel'fand-Dikii hierarchy was introduced by Nijhoff, Papageorgiou, Capel and Quispel in 1992 as the family of partial difference equations generalizing to higher rank the lattice Korteweg-de Vries systems, and includes in particular the lattice Boussinesq system. We present a Lagrangian for the generic member of the lattice Gel'fand-Dikii hierarchy, and show that it can be considered as a Lagrangian 2-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus the multiform structure proposed in arXiv:0903.4086v2 [nlin.SI] is extended to a multi-component system., Comment: 12 pages

We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation of the form of the nonlinear Schrodinger (NLS) equation. If the starting lattice equation is integrable then the resulting NLS equation turns out to be integrable, while if the starting equation is linearizable we get a linear Schrodinger equation. On the other hand, if we start with a non-integrable lattice equation we may obtain a non-integrable NLS equation. This conjecture is confirmed by many examples., 12 pages

In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain the integrable deautonomisations of the equations examined. In the case of linearisable lattice equations we show that the degree growth is slower than in the case of equations integrable through Inverse Scattering Transform techniques.

We study the integrability of mappings obtained as reductions of the discrete Korteweg–de Vries (KdV) equation and of two copies of the discrete potential KdV (pKdV) equation. We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville–Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.

Bilinear forms for some nonlinear partial difference equations(discrete soliton equations) are derived based on the results of singularity confinement. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed., Comment: 14 pages, LaTex

We present a discretization of the Jacobi last multiplier, with some applications to the computation of solutions of linear partial difference equations.

In this paper, a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations (PΔEs). The identified PΔEs can be classified into QRT type, KdV type, mKdV type and sG type. The integrability of the identified partial difference equations is also discussed.

We propose a method to identify and examine every partial difference Lax pair that consists of two 2×2 discrete linear problems, where the matrices contain one separable term in each entry. We thereby derive new, higher order versions of the lattice sine-Gordon and lattice modified Korteweg–de Vries equations, while showing that there can be no other partial difference equations associated with this type of Lax pair.

We examine two integrable discrete lattice equations obtained by Levi and Yamilov. We show that the first one is a form of the lattice KdV equation already obtained by Hirota and Tsujimoto, while the second one is a discrete form of mKdV. We present the Miura transformations between the various equations involved, including the more familiar potential form of the lattice mKdV.

We give a method to calculate closed-form expressions in terms of multi-sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations. Important ingredients are the staircase method, a non-commutative Vieta formula and certain splittings of the Lax matrices. The method is applied to all equations of the Adler–Bobenko–Suris classification, with the exception of Q4.

A more elaborate discrete multiple scales analysis than that used by Newell in 1977 is performed on the Zabusky-Kruskal discretization of the Korteweg-de Vries (KdV) equation. This eventually leads to a set of partial difference equations describing the modulational behavior of a small harmonic wave modulated by a slowly varying envelope. In the case of certain modes of the carrier wave, the multiple scales analysis breaks down, indicating that in these cases the numerical solution deviates in behavior from that of the KdV equation. Numerical experiments are reported which confirm this.

Using the method of difference equations the bound states of excitations in a linear chain are investigated. It is possible to obtain solutions of the partial difference equations and the energy of “biphonons” by a separation ansatz. The method of power series expansion is not so successful. The spectrum of the states with two phonons in two coupled linear chains (as a model of the system film-substrate) is investigated additionally. Mit Hilfe der Methode der Differenzengleichungen werden gebundene Zustande von Anregungen in einer linearen Kette untersucht. Durch einen Separationsansatz gelingt es, die entsprechenden partiellen Differenzengleichungen zu losen und die Energie der „Biphononen” zu berechnen. Mit Hilfe der Methode der Potenzreihen-Entwicklung ist das nicht gelungen. Anschliesend wird das Spektrum der Zwei-Phononen-Zustande in zwei gekoppelten linearen Ketten (als Modell fur das System Schicht–Substrate) untersucht.

Mass equations M(N,Z) are generally obtained by establishing the analytical form of the equation on the basis of nuclear structure considerations and by subsequently minimizing the differences M(N,Z) — Mexp(N,Z) for all known masses by adjusting parameters. An alternate approach is described in this contribution. It makes use of nuclidic mass relationships which represent partial difference equations. Some aspects of this approach have been reported earlier[1,2].

For all temperatures, the two-point function of the two-dimensional Ising model is shown to be expressible in terms of the solution of a non-linear partial difference equation on the lattice. From this difference equation, the known results for the two-point function of the Ising field theory are regained by taking the scaling limit.

Several explicit numerical solutions of generalized nuclidic mass relationships (partial difference equations) have been derived. These solutions are mass equations with about 230 parameters which reproduce more than 1000 experimental masses with N, Z ≧ 20 with a standard deviation of about 300 keV. The internal consistency of the solutions and other aspects such as the ability to describe the experimental Coulomb displacement energies are explored. The solutions are compared to the transverse and longitudinal mass equations of Garvey and Kelson and the shell-model mass equations of Liran and Zeldes for their reliability to predict masses of very neutron-rich and proton-rich nuclei.

Quadratic identities for Ising model correlations on a general planar lattice are derived. For the ordinary Ising model, they imply the nonlinear partial difference equations of McCoy and Wu. At the critical temperature Hirota's discrete-time Toda equation is found.

Any mass equation M(N,Z) may be expressed as a special solution of an inhomogeneous partial difference equation. One only has to introduce an appropriate difference operator D and calculate κ(N,Z) = D M(N,Z). The inhomogeneous partial difference equation $${\text{D M}}\left( {{\text{N}},{\text{Z}}} \right){\text{ }} = {\text{ }}\kappa \left( {{\text{N}},{\text{Z}}} \right)$$ (1) contains the original mass equation as a special solution. The most general solution of eq. (1) will include solutions of the homogeneous equation which can, for example, be used to generate shell correction terms.