22,769 research outputs found
An integrable semi-discretization of the Camassa-Holm equation and its determinant solution
An integrable semi-discretization of the Camassa-Holm equation is presented.
The keys of its construction are bilinear forms and determinant structure of
solutions of the CH equation. Determinant formulas of -soliton solutions of
the continuous and semi-discrete Camassa-Holm equations are presented. Based on
determinant formulas, we can generate multi-soliton, multi-cuspon and
multi-soliton-cuspon solutions. Numerical computations using the integrable
semi-discrete Camassa-Holm equation are performed. It is shown that the
integrable semi-discrete Camassa-Holm equation gives very accurate numerical
results even in the cases of cuspon-cuspon and soliton-cuspon interactions. The
numerical computation for an initial value condition, which is not an exact
solution, is also presented
Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schr\"odinger Equation
It is shown that the -dark soliton solutions of the integrable discrete
nonlinear Schr\"odinger (IDNLS) equation are given in terms of the Casorati
determinant. The conditions for reduction, complex conjugacy and regularity for
the Casorati determinant solution are also given explicitly. The relationship
between the IDNLS and the relativistic Toda lattice is discussed.Comment: First version was uploaded in 23 Jun 2005. Published in Journal of
the Physical Society of Japan in May, 200
Casorati Determinant Solution for the Relativistic Toda Lattice Equation
The relativistic Toda lattice equation is decomposed into three Toda systems,
the Toda lattice itself, B\"acklund transformation of Toda lattice and discrete
time Toda lattice. It is shown that the solutions of the equation are given in
terms of the Casorati determinant. By using the Casoratian technique, the
bilinear equations of Toda systems are reduced to the Laplace expansion form
for determinants. The -soliton solution is explicitly constructed in the
form of the Casorati determinant.Comment: 19 pages in plain Te
The Andrews-Gordon identities and -multinomial coefficients
We prove polynomial boson-fermion identities for the generating function of
the number of partitions of of the form , with
, and . The bosonic side of
the identities involves -deformations of the coefficients of in the
expansion of . A combinatorial interpretation for these
-multinomial coefficients is given using Durfee dissection partitions. The
fermionic side of the polynomial identities arises as the partition function of
a one-dimensional lattice-gas of fermionic particles. In the limit
, our identities reproduce the analytic form of Gordon's
generalization of the Rogers--Ramanujan identities, as found by Andrews. Using
the duality, identities are obtained for branching functions
corresponding to cosets of type of fractional level .Comment: 31 pages, Latex, 9 Postscript figure
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