Soliton solutions for ABS lattice equations: I. Cauchy matrix approach

@article{Nijhoff2009SolitonSF,
  title={Soliton solutions for ABS lattice equations: I. Cauchy matrix approach},
  author={Frank W Nijhoff and James Atkinson and Jarmo Hietarinta},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2009},
  volume={42},
  pages={404005}
}
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… 

Elliptic N-soliton Solutions of ABS Lattice Equations

Elliptic N-soliton-type solutions, that is, solutions emerging from the application of N consecutive Backlund transformations to an elliptic seed solution, are constructed for all equations in the

Elliptic Solutions of ABS Lattice Equations

Elliptic N-soliton-type solutions, i.e. solutions emerging from the application of N consecutive B\"acklund transformations to an elliptic seed solution, are constructed for all equations in the ABS

Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization

In Part I soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by

A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, and

Multidimensional inverse scattering of integrable lattice equations

We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of

A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations

  • Mats Vermeeren
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2019
A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the

Singular-Boundary Reductions of Type-Q ABS Equations

We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2, and Q1). Singular boundary problems for these equations are systematized in

Spectrum transformation and conservation laws of lattice potential KdV equation

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we

Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity

We establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for

An inverse scattering transform for the lattice potential KdV equation

The lattice potential Korteweg–de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated
...

References

SHOWING 1-10 OF 38 REFERENCES

Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization

In Part I soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by

Seed and soliton solutions for Adler's lattice equation

Adler's lattice equation has acquired the status of a master equation among 2D discrete integrable systems. In this paper we derive what we believe are the first explicit solutions of this equation.

The direct method in soliton theory

The bilinear, or Hirota's direct, method was invented in the early 1970s as an elementary means of constructing soliton solutions that avoided the use of the heavy machinery of the inverse scattering

Integrable mappings derived from soliton equations

Soliton solutions for Q3

We construct N-soliton solutions to the equation called Q3 in the recent Adler–Bobenko–Suris classification. An essential ingredient in the construction is the relationship of (Q3)δ=0 to the equation

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the

Q4: integrable master equation related to an elliptic curve

One of the most fascinating and technically demanding parts of the theory of two-dimensional integrable systems constitutes the models with the spectral parameter on an elliptic curve, including

Symmetries and Integrability of Difference Equations

The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several

Discrete nonlinear hyperbolic equations. Classification of integrable cases

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ℤ2. The fields are associated with the vertices and an equation of the form Q(x1, x2, x3, x4) = 0 relates four

Solutions of Adler’s Lattice Equation Associated with 2-Cycles of the Bäcklund Transformation

Abstract The Bäcklund transformation (BT) of Adler’s lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is