1,720,977 research outputs found
A discrete linearizability test based on multiscale analysis
Full text linkIn 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 Möbius transformation
Multiscale expansion and integrability properties of the lattice potential KdV equation
Full text linkWe apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schrödinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schrödinger equation
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
Full text linkIn this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the A_1, A_2 and A_3 linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable
Integrability test of discrete nonlinear Schrodinger equations via multiscale reduction
No full textIn this article, we consider the multiscale reduction around the harmonic solution of a general discrete nonlinear Schrodinger equation (dNLSE) depending on constant coefficients. According to the values of the coefficients we can have both integrable and non-integrable dNLSEs. For all values of the coefficients entering the dNLSE, non-secularity conditions provide an integrable NLSE at the lowest order in the perturbation parameter. However at higher order in the perturbation expansion the request that the expansion is compatible with the NLSE hierarchy gives integrability conditions which are not satisfied for the non-integrable dNLSEs
The lattice Schwarzian KdV equation and its symmetries
No full textIn 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
Reconstructing a lattice equation: A non-autonomous approach to the Hietarinta equation
Full text linkIn this paper we construct a non-autonomous version of the Hietarinta equation [Hietarinta J., J. Phys. A: Math. Gen. 37 (2004), L67-L73] and study its integrability properties. We show that this equation possess linear growth of the degrees of iterates, generalized symmetries depending on arbitrary functions, and that it is Darboux integrable. We use the first integrals to provide a general solution of this equation. In particular we show that this equation is a sub-case of the non-autonomous QV equation, and we provide a nonautonomous Möbius transformation to another equation found in [Hietarinta J., J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 223-230] and appearing also in Boll’s classification [Boll R., Ph.D. Thesis, Technische Universität Berlin, 2012]
Multiscale reduction of discrete nonlinear Schrodinger equations
No full textWe use a discrete multiscale analysis to study the asymptotic integrability of differential-difference equations. In particular, we show that multiscale perturbation techniques provide an analytic tool to derive necessary integrability conditions for two well-known discretizations of the nonlinear Schrodinger equation
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