arXiv subjects
Publications and source records attributed to A. Ramani.
We examine a family of 3-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through the application of the singularity confinement criterion and are explicitly integrated.
We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete form of this equation, the Gambier mapping, and present conditions for its integrability. Finally, we obtain the reductions of the Gambier mapping, identify their integrable forms and compute their continuous limits.
We present the bilinear forms of the (continuous) Painlevé equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new and interesting result is the bilinearization of the P$_{\rm VI}$ equation, something that was missing till now.
Starting from integrable cellular automata we present a novel form of Painlevé equations. These equations are discrete in both the independent variable and the dependent one. We show that they capture the essence of the behavior of the Painlevé equations organizing themselves into a coalescence cascade and possessing special solutions. A necessary condition for the integrability of cellular automata is also presented.
We show that the recently derived ($q$-) discrete form of the Painlevé VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.
Starting from the standard form of the five discrete Painlevé equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painlevé equations. A particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new $n$-dependence and introduces more new discrete Painlevé equations.
We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained through a system of Riccati equations. The same holds true in the discrete case also. We use the singularity confinement criterion in order to study the integrability of this new mapping.
By analogy to the continuous Painlevé II equation, we present particular solutions of the discrete Painlevé II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $τ$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.
In a recent publication we proposed an extension of Hirota's bilinear formalism to arbitrary multilinearities. The trilinear (and higher) operators were constructed from the requirement of gauge invariance for the nonlinear equation. Here we concentrate on the trilinear case, and use singularity analysis in order to single out equations that are likely to be integrable. New PDE's are thus obtained, along with others already well-known for their integrability and for which we obtain here the trilinear expression. To appear in the proceedings of NEEDS'94 (11-18 September, Los Alamos)
We introduce multilinear operators, that generalize Hirota's bilinear $D$ operator, based on the principle of gauge invariance of the $τ$ functions. We show that these operators can be constructed systematically using the bilinear $D$'s as building blocks. We concentrate in particular on the trilinear case and study the possible integrability of equations with one dependent variable. The 5th order equation of the Lax-hierarchy as well as Satsuma's lowest-order gauge invariant equation are shown to have simple trilinear expressions. The formalism can be extended to an arbitrary degree of multilinearity.
We present an integrability criterion for rational mappings based on two requirements. First, that a given point should have a unique preimage under the mapping and, second, that the spontaneously appearing singularities be confined to a few iteration steps. We present several examples of known integrable mappings that meet these requirements and, also, use our algorithm in order to derive new examples of integrable mappings.
We discuss the freedom in the background field (vacuum) on top of which the solitons are built. If the Hirota bilinear form of a soliton equation is given by $A(D_{\vec x})\bd GF=0,\, B(D_{\vec x})(\bd FF - \bd GG)=0$ where both $A$ and $B$ are even polynomials in their variables, then there can be a continuum of vacua, parametrized by a vacuum angle $ϕ$. The ramifications of this freedom on the construction of one- and two-soliton solutions are discussed. We find, e.g., that once the angle $ϕ$ is fixed and we choose $u=\arctan G/F$ as the physical quantity, then there are four different solitons (or kinks) connecting the vacuum angles $\pmϕ$, $\pmϕ\pm\Pi2$ (defined modulo $π$). The most interesting result is the existence of a ``ghost'' soliton; it goes over to the vacuum in isolation, but interacts with ``normal'' solitons by giving them a finite phase shift.
We present particular solutions of the discrete Painlevé III (d-P$\rm_{III}$) equation of rational and special function (Bessel) type. These solutions allow us to establish a close parallel between this discrete equation and its continuous counterpart. Moreover, we propose an alternate form for d-P$\rm_{III}$ and confirm its integrability by explicitly deriving its Lax pair.