1,720,966 research outputs found
Three-point partial difference equations linearizable by local and nonlocal transformations
No full text"\"We consider a class of nonlinear partial difference equations defined on three points of a plane lattice. We construct conditions for this class of partial difference equations to be linearizable through a point or a Cole–Hopf transformation. Using these conditions we are able to classify all multilinear linearizable equations belonging to this class\"
Linearization through symmetries for discrete equations
No full text"We show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete Cole–Hopf transformation. If the equation is linearizable, then the symmetry provides the linearizing transformation. At the end, we present a few examples of applications for equations defined on four lattice points
Classification of discrete equations linearizable by point transformation on a square lattice
No full text"We provide a complete set of linearizability conditions for nonlinear partial difference equations defined on four points and, using them, we classify all linearizable multilinear partial difference equations defined on four points up to a Mobious transformation
Classification of multilinear real quadratic partial difference equations linearizable by point and Hopf-Cole transformations
No full text""We use the conditions of linearizability by point and by Hopf–Cole transformations, introduced recently by the authors, to classify all real multilinear equations on a square lattice with at most quadratic nonlinearity. We find that, up to a linear transformation of the dependent variable and an exchange of the independent variables, only two equations in this class are linearizable by point transformations and none by Hopf–Cole transformations."
Four Points Linearizable Lattice Schemes
No full text"We provide conditions for a lattice scheme defined on a four points lattice to be linearizable by a point transformation. We apply the obtained conditions to a symmetry preserving difference scheme for the Burgers potential introduced by Dorodnitsyn and show that it is not linearizable
Algebraic entropy, symmetries and linearization of quad equations consistent on the cube
No full textWe 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
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
Contact transformations for difference schemes
No full text""We define a class of transformations of the dependent and independent variables in an ordinary difference scheme. The transformations leave the solution set of the system invariant and reduces to a group of contact transformations in the continuous limit. We use a simple example to show that the class is not empty and that such 'contact transformations for discrete systems' genuinely exist"
Classification of discrete systems on a square lattice
Full text linkWe consider the classification up to a Möbius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice
On partial differential and difference equations with symmetries depending on arbitrary functions
Full text linkIn 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
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