102 research outputs found
Transverse instability and its long-term development for solitary waves of the (2+1)-Boussinesq equation
The stability properties of line solitary wave solutions of the
(2+1)-dimensional Boussinesq equation with respect to transverse perturbations
and their consequences are considered. A geometric condition arising from a
multi-symplectic formulation of this equation gives an explicit relation
between the parameters for transverse instability when the transverse
wavenumber is small. The Evans function is then computed explicitly, giving the
eigenvalues for transverse instability for all transverse wavenumbers. To
determine the nonlinear and long time implications of transverse instability,
numerical simulations are performed using pseudospectral discretization. The
numerics confirm the analytic results, and in all cases studied, transverse
instability leads to collapse.Comment: 16 pages, 8 figures; submitted to Phys. Rev.
Integrable Time-Discretisation of the Ruijsenaars-Schneider Model
An exactly integrable symplectic correspondence is derived which in a
continuum limit leads to the equations of motion of the relativistic
generalization of the Calogero-Moser system, that was introduced for the first
time by Ruijsenaars and Schneider. For the discrete-time model the equations of
motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2
Heisenberg magnet. We present a Lax pair, the symplectic structure and prove
the involutivity of the invariants. Exact solutions are investigated in the
rational and hyperbolic (trigonometric) limits of the system that is given in
terms of elliptic functions. These solutions are connected with discrete
soliton equations. The results obtained allow us to consider the Bethe Ansatz
equations as ones giving an integrable symplectic correspondence mixing the
parameters of the quantum integrable system and the parameters of the
corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st
Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
New extensions of the KP and modified KP hierarchies with self-consistent
sources are proposed. The latter provide new generalizations of
-dimensional integrable equations, including the DS-III equation and the
-wave problem. Furthermore, we recover a system that contains two types of
the KP equation with self-consistent sources as special cases. Darboux and
binary Darboux transformations are applied to generate solutions of the
proposed hierarchies
The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces
The Hamiltonian representation for the hierarchy of Lax-type flows on a dual
space to the Lie algebra of shift operators coupled with suitable
eigenfunctions and adjoint eigenfunctions evolutions of associated spectral
problems is found by means of a specially constructed Backlund transformation.
The Hamiltonian description for the corresponding set of squared eigenfunction
symmetry hierarchies is represented. The relation of these hierarchies with Lax
integrable (2+1)-dimensional differential-difference systems and their triple
Lax-type linearizations is analysed. The existence problem of a Hamiltonian
representation for the coupled Lax-type hierarchy on a dual space to the
central extension of the shift operator Lie algebra is solved also
On the numerical evaluation of algebro-geometric solutions to integrable equations
Physically meaningful periodic solutions to certain integrable partial
differential equations are given in terms of multi-dimensional theta functions
associated to real Riemann surfaces. Typical analytical problems in the
numerical evaluation of these solutions are studied. In the case of
hyperelliptic surfaces efficient algorithms exist even for almost degenerate
surfaces. This allows the numerical study of solitonic limits. For general real
Riemann surfaces, the choice of a homology basis adapted to the
anti-holomorphic involution is important for a convenient formulation of the
solutions and smoothness conditions. Since existing algorithms for algebraic
curves produce a homology basis not related to automorphisms of the curve, we
study symplectic transformations to an adapted basis and give explicit formulae
for M-curves. As examples we discuss solutions of the Davey-Stewartson and the
multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure
Dressing method and the coupled KP hierarchy
The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by
using the dressing method. It is shown that the coupled KP hierarchy can be
reformulated as a reduced case of the 2-component KP hierarchy.Comment: 11 pages, LaTeX, no figure
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