314 research outputs found
Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
We perform a classification of integrable systems of mixed scalar and vector
evolution equations with respect to higher symmetries. We consider polynomial
systems that are homogeneous under a suitable weighting of variables. This
paper deals with the KdV weighting, the Burgers (or potential KdV or modified
KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings.
The case of other weightings will be studied in a subsequent paper. Making an
ansatz for undetermined coefficients and using a computer package for solving
bilinear algebraic systems, we give the complete lists of 2nd order systems
with a 3rd order or a 4th order symmetry and 3rd order systems with a 5th order
symmetry. For all but a few systems in the lists, we show that the system (or,
at least a subsystem of it) admits either a Lax representation or a linearizing
transformation. A thorough comparison with recent work of Foursov and Olver is
made.Comment: 60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus
several minor changes, to appear in J.Phys.
Integrable Quartic Potentials and Coupled KdV Equations
We show a surprising connection between known integrable Hamiltonian systems
with quartic potential and the stationary flows of some coupled KdV systems
related to fourth order Lax operators. In particular, we present a connection
between the Hirota-Satsuma coupled KdV system and (a generalisation of) the
integrable case quartic potential. A generalisation of the case
is similarly related to a different (but gauge related) fourth order Lax
operator. We exploit this connection to derive a Lax representation for each of
these integrable systems. In this context a canonical transformation is derived
through a gauge transformation.Comment: LaTex, 11 page
Constructing a Supersymmetric Integrable System from the Hirota Method in Superspace
An N=1 supersymmetric system is constructed and its integrability is shown by
obtaining three soliton solutions for it using the supersymmetric version of
Hirota's direct method.Comment: 10 pages, no figure
On application of Liouville type equations to constructing B\"acklund transformations
It is shown how pseudoconstants of the Liouville-type equations can be
exploited as a tool for construction of the B\"acklund transformations. Several
new examples of such transformations are found. In particular we obtained the
B\"acklund transformations for a pair of three-component analogs of the
dispersive water wave system, and auto-B\"acklund transformations for coupled
three-component KdV-type systems.Comment: 11 pages, no figure
Note on Nonlinear Schr\"odinger Equation, KdV Equation and 2D Topological Yang-Mills-Higgs Theory
In this paper we discuss the relation between the (1+1)D nonlinear
Schr\"odinger equation and the KdV equation. By applying the boson/vortex
duality, we can map the classical nonlinear Schr\"odinger equation into the
classical KdV equation in the small coupling limit, which corresponds to the UV
regime of the theory. At quantum level, the two theories satisfy the Bethe
Ansatz equations of the spin- XXX chain and the XXZ chain in the
continuum limit respectively. Combining these relations with the dualities
discussed previously in the literature, we propose a duality web in the UV
regime among the nonlinear Schr\"odinger equation, the KdV equation and the 2D
topological Yang-Mills-Higgs theory.Comment: 20 pages, 1 figure; V2: typos correcte
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