Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura and Yasuhiro Ohta
Journal of Physics A: Theoretical and Mathematical 45 045206 Jan 2012 [Refereed]
We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified Kd...
Journal of Physics A: Mathematica and Theoretical 44 395201 Sep 2011 [Refereed]
We consider integrable discretizations of some soliton equations associated
with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam
equation, the complex Dym equation, and the short pulse equation. They are
related to the modified KdV or the sine-Gordon equations by the hodograph
transformations. Based on the observation that the hodograph transformations
are regarded as the Eu...
Journal of Math-for-Industry 3(2011A) 53-62 Apr 2011 [Refereed]
Various solutions to the discrete Schwarzian KdV equation are discussed. We
first derive the bilinear difference equations of Hirota type of the discrete
Schwarzian KP equation, which is decomposed into three discrete two-dimensional
Toda lattice equations. We then construct two kinds of solutions in terms of
the Casorati determinant. We derive the discrete Schwarzian KdV equation on an
inhomog...
Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda
May 2011
We present an explicit formula for the discrete power function introduced by
Bobenko, which is expressed in terms of the hypergeometric functions for
the sixth Painlevé equation. The original definition of the discrete power
function imposes strict conditions on the domain and the value of the exponent.
However, we show that one can extend the value of the exponent to arbitrary
complex...
Kyushu Journal of Mathematics In press 2011 [Refereed]
We construct explicit solutions to discrete motion of discrete plane curves
which has been introduced by one of the authors recently. Explicit formulas in
terms the function are presented. Transformation theory of motions of
both smooth and discrete curves is developed simultaneously.
We consider an integrability test for ultradiscrete equations based on the
singularity confinement analysis for discrete equations. We show how
singularity pattern of the test is transformed into that of ultradiscrete
equation. The ultradiscrete solution pattern can be interpreted as a perturbed
solution. We can also check an integrability of a given equation by a
perturbation growth of a solut...
Kenji Kajiwara, Yasuhiro Ohta, Junkichi Satsuma, Basil Grammaticos, Alfred Ramani
Oct 1993
We present a class of solutions to the discrete Painlevé-II equation for
particular values of its parameters. It is shown that these solutions can be
expressed in terms of Casorati determinants whose entries are discrete Airy
functions. The analogy between the function for the discrete P and the that of the discrete Toda molecule equation is pointed out.
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