249 research outputs found
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 Backlund transformation, and the method of solving a
Riccati map by exploiting two kn own particular solutions. This leads to an
expression for the N-soliton-type solutions of a generic equation within this
class. As a particular instance we give an explicit N-soliton solution for the
primary model, which is Adler's lattice equation (or Q4).Comment: 22 page
Classification of 3-dimensional integrable scalar discrete equations
We classify all integrable 3-dimensional scalar discrete quasilinear
equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An
equation Q=0 is called integrable if it may be consistently imposed on all
3-dimensional elementary faces of the 4-dimensional lattice.
Under the natural requirement of invariance of the equation under the action
of the complete group of symmetries of the cube we prove that the only
nontrivial (non-linearizable) integrable equation from this class is the
well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2
has been made.) (Version 3: A few small corrections: one more reference added,
the main statement stated more explicitly.)Comment: 20 p. LaTeX + 1 EPS figur
On the structure of the B\"acklund transformations for the relativistic lattices
The B\"acklund transformations for the relativistic lattices of the Toda type
and their discrete analogues can be obtained as the composition of two duality
transformations. The condition of invariance under this composition allows to
distinguish effectively the integrable cases. Iterations of the B\"acklund
transformations can be described in the terms of nonrelativistic lattices of
the Toda type. Several multifield generalizations are presented
Cauchy problem for integrable discrete equations on quad-graphs
Initial value problems for the integrable discrete equations on quadgraphs
are investigated. We give a geometric criterion of when such a
problem is well-posed. In the basic example of the discrete KdV equation
an effective integration scheme based on the matrix factorization
problem is proposed and the interaction of the solutions with the localized
defects in the regular square lattice are discussed in details.
The examples of kinks and solitons on various quad-graphs, including
quasiperiodic tilings, are presented
On the one class of hyperbolic systems
The classification problem is solved for some type of nonlinear lattices. These lattices are closely related to the lattices of Ruijsenaars-Toda type and define the Bäcklund auto-transformations for the class of two-component hyperbolic systems
Asymptotic solitons of the Johnson equation
We prove the existence of non-decaying real solutions of the Johnson
equation, vanishing as . We obtain asymptotic formulas as
for the solutions in the form of an infinite series of asymptotic
solitons with curved lines of constant phase and varying amplitude and width
On discrete integrable equations with convex variational principles
We investigate the variational structure of discrete Laplace-type equations
that are motivated by discrete integrable quad-equations. In particular, we
explain why the reality conditions we consider should be all that are
reasonable, and we derive sufficient conditions (that are often necessary) on
the labeling of the edges under which the corresponding generalized discrete
action functional is convex. Convexity is an essential tool to discuss
existence and uniqueness of solutions to Dirichlet boundary value problems.
Furthermore, we study which combinatorial data allow convex action functionals
of discrete Laplace-type equations that are actually induced by discrete
integrable quad-equations, and we present how the equations and functionals
corresponding to (Q3) are related to circle patterns.Comment: 39 pages, 8 figures. Revision of the whole manuscript, reorder of
sections. Major changes due to additional reality conditions for (Q3) and
(Q4): new Section 2.3; Theorem 1 and Sections 3.5, 3.6, 3.7 update
Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations we derive new vector Yang-Baxter map and integrable discrete vector sine-Gordon equation on a sphere
Characteristics of Conservation Laws for Difference Equations
Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation
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