Christian Scimiterna

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the dis- crete Schrodinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Backlund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Backlund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

The Kundu-Eckhaus equation and its discretizations

In this paper we show that the complex Burgers and the Kundu–Eckhaus equations are related by a Miura transformation. We use this relation to discretize the Kundu–Eckhaus equation.

The lattice Schwarzian KdV equation and its symmetries

In this paper, we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (1SKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the 1SKdV equation to construct non-autonomous non-integrable generalized symmetries.

ON PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS WITH SYMMETRIES DEPENDING ON ARBITRARY FUNCTIONS

In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial difference equations when 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.

On the integrability of a new lattice equation found by multiple scale analysis

In this paper we discuss the integrability properties of a nonlinear partial difference equation on the square obtained by the multiple scale integrability test from a class of multilinear dispersive equations defined on a four-point lattice.

Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf{Cole Transformations ?

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.

Multiscale expansion of the lattice potential KdV equation on functions of an infinite slow-varyness order

We present a discrete multiscale expansion of the lattice potential Korteweg–de Vries (lpKdV) equation on functions of an infinite order of slow varyness. To do so, we introduce a formal expansion of the shift operator on many lattices holding at all orders. The lowest secularity condition from the expansion of the lpKdV equation gives a nonlinear lattice equation, depending on shifts of all orders, of the form of the nonlinear Schrodinger equation.

A discrete linearizability test based on multiscale analysis

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 A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Mobius transformation.

Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

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 Bäcklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the H4 trapezoidal and the H6 families are linearizable and in a few examples we show how we can effectively linearize them.

Multiscale expansion on the lattice and integrability of partial difference equations

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.