1,720,967 research outputs found
Non-point integrable symmetries for equations on the lattice
No full textWe present a new class of non-point groups of transformations for scalar evolution chain equations. Then we construct the class of differential equations on the lattice which admits such group transformations
On the integrability of a new discrete nonlinear Schrodinger equation
No full textWe consider the nonlinear Schrodinger equation on the lattice introduced by Leon and Manna two years ago to describe the slowly varying envelope approximation of some nonlinear differential difference equations. We show that this equation does not admit local generalized symmetries of order greater than three. In such a way we prove that the Leon and Manna discrete nonlinear Schrodinger equation does not have the same integrability properties as the Toda lattice equation, from which it has been derived. At the end we provide some reasoning to justify the result obtained
Conditions for the existence of higher symmetries of evolutionary equations on the lattice
No full textIn this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential-difference equations depending only on nearest neighboring interaction. These conditions are applied to prove the existence of new integrable equations belonging to this class, (C) 1997 American Institute of Physics
Darboux integrability of trapezoidal H 4 and H 4 families of lattice equations I: First integrals
No full textIn this paper we prove that the trapezoidal H4 and the H6 families of quadequations are Darboux integrable by constructing their first integrals. This result explains why the rate of growth of the degrees of the iterates of these equations is linear (Gubbiotti et al 2016 J. Nonlinear Math. Phys. 23 507-43), which according to the algebraic entropy conjecture implies linearizability. We conclude by showing how first integrals can be used to obtain general solutions
Dilation symmetries and equations on the lattice
Full text linkWe discuss the role of dilation symmetries for differential difference equations depending on nearest-neighbour interactions. In particular, we show that for a simple class of differential difference equations of this kind, symmetries which depend linearly on lime are only compatible with linearizable equations
Darboux integrability of trapezoidal H4 and H6 families of lattice equations II: General solutions
Full text linkIn this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal H4 equations and the H6 equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved
On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations
No full textWe construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete 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
Integrability conditions for n and t dependent dynamical lattice equations
No full textConditions necessary for the existence of local higher order generalized symmetries and conservation laws are derived for a class of dynamical lattice equations with explicit dependence on the spatial discrete variable and on time. We explain how to use the obtained conditions for checking a given equation. We apply those conditions to the study of a special class of differential difference equations interesting from the physical point of view
Non-invertible transformations of differential-difference equations
No full textWe discuss aspects of the theory of non-invertible transformations of differential-difference equations and, in particular, the notion of Miura type transformation. We introduce the concept of non-Miura type linearizable transformation and we present techniques that allow one to construct simple linearizable transformations and might help one to solve classification problems. This theory is illustrated by the example of a new integrable differential-difference equation depending on five lattice points, interesting from the viewpoint of the non-invertible transformation, which relate it to an Itoh-Narita-Bogoyavlensky equation
On Miura transformations and Volterra-type equations
No full textWe construct Miura transformations mapping the scalarspectral problems of the integrable lattice equations belonging tothe Adler-Bobenko-Suris (ABS) list into the discrete Schrodingerspectral problem associated with Volterra-type equations. We showthat the ABS equations correspond to Backlund transformations forsome particular cases of the discrete Krichever-Novikov equationfound by Yamilov (YdKN equation). This enables us to construct newgeneralized symmetries for the ABS equations. The same can be saidabout the generalizations of the ABS equations introduced by Tongas,Tsoubelis and Xenitidis. All of them generate Backlundtransformations for the YdKN equation. The higher order generalizedsymmetries we construct in the present paper confirm theirintegrability
- …
