62 research outputs found
On a Lagrangian reduction and a deformation of completely integrable systems
We develop a theory of Lagrangian reduction on loop groups for completely
integrable systems after having exchanged the role of the space and time
variables in the multi-time interpretation of integrable hierarchies. We then
insert the Sobolev norm in the Lagrangian and derive a deformation of the
corresponding hierarchies. The integrability of the deformed equations is
altered and a notion of weak integrability is introduced. We implement this
scheme in the AKNS and SO(3) hierarchies and obtain known and new equations.
Among them we found two important equations, the Camassa-Holm equation, viewed
as a deformation of the KdV equation, and a deformation of the NLS equation
A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion
We propose a new method for discretizing the time variable in integrable
lattice systems while maintaining the locality of the equations of motion. The
method is based on the zero-curvature (Lax pair) representation and the
lowest-order "conservation laws". In contrast to the pioneering work of
Ablowitz and Ladik, our method allows the auxiliary dependent variables
appearing in the stage of time discretization to be expressed locally in terms
of the original dependent variables. The time-discretized lattice systems have
the same set of conserved quantities and the same structures of the solutions
as the continuous-time lattice systems; only the time evolution of the
parameters in the solutions that correspond to the angle variables is
discretized. The effectiveness of our method is illustrated using examples such
as the Toda lattice, the Volterra lattice, the modified Volterra lattice, the
Ablowitz-Ladik lattice (an integrable semi-discrete nonlinear Schroedinger
system), and the lattice Heisenberg ferromagnet model. For the Volterra lattice
and modified Volterra lattice, we also present their ultradiscrete analogues.Comment: 61 pages; (v2)(v3) many minor correction
Complete integrability of shock clustering and Burgers turbulence
We consider scalar conservation laws with convex flux and random initial
data. The Hopf-Lax formula induces a deterministic evolution of the law of the
initial data. In a recent article, we derived a kinetic theory and Lax
equations to describe the evolution of the law under the assumption that the
initial data is a spectrally negative Markov process. Here we show that: (i)
the Lax equations are Hamiltonian and describe a principle of least action on
the Markov group that is in analogy with geodesic flow on ; (ii) the Lax
equations are completely integrable and linearized via a loop-group
factorization of operators; (iii) the associated zero-curvature equations can
be solved via inverse scattering. Our results are rigorous for -dimensional
approximations of the Lax equations, and yield formulas for the limit . The main observation is that the Lax equations are a
limit of a Markovian variant of the -wave model. This allows us to introduce
a variety of methods from the theory of integrable systems
Higher dimensional integrable deformations of the modified KdV equation
The derivation of nonlinear integrable evolution partial differential
equations in higher dimensions has always been the holy grail in the field of
integrability. The well-known modified KdV equation is a prototypical example
of integrable evolution equations in one spatial dimension. Do there exist
integrable analogs of modified KdV equation in higher spatial dimensions? In
what follows, we present a positive answer to this question. In particular,
rewriting the (1+1)-dimensional integrable modified KdV equation in
conservation forms and adding deformation mappings during the process allow one
to construct higher dimensional integrable equations. Further, we illustrate
this idea with examples from the modified KdV hierarchy, also present the Lax
pairs of these higher dimensional integrable evolution equations.Comment: 7 pages, 3 figure
Sasa-Satsuma hierarchy of integrable evolution equations
We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth-order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly.The authors gratefully acknowledge the support of the Australian Research Council (Discovery Projects
DP140100265 and DP150102057) and support from the Volkswagen Stiftung. N.A. is a recipient of the Alexander von Humboldt Award. U.B. acknowledges support by the German Research Foundation in the framework of the Collaborative Research Center 787 âSemiconductor Nanophotonicsâ under project B5. Sh.A. acknowledges support of the German Research Foundation under Project No. 389251150
Sasa--Satsuma hierarchy of integrable evolution equations
We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth- order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly
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