109 research outputs found
Constrained lattice-field hierarchies and Toda system with Block symmetry
In this paper, we construct the additional -symmetry and ghost symmetry of
two-lattice field integrable hierarchies. Using the symmetry constraint, we
construct constrained two-lattice integrable systems which contain several new
integrable difference equations. Under a further reduction, the constrained
two-lattice integrable systems can be combined into one single integrable
system, namely the well-known one dimensional original Toda hierarchy. We prove
that the one dimensional original Toda hierarchy has a nice Block Lie symmetry.Comment: 16 Pages, accepted for publication in International Journal of
Geometric Methods in Modern Physic
Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries
In this article, we will construct the additional perturbative quantum torus
symmetry of the dispersionless BKP hierarchy basing on the
infinite dimensional Lie symmetry. These results show that the complete quantum
torus symmetry is broken from the BKP hierarchy to its dispersionless
hierarchy. Further a series of additional flows of the multicomponent BKP
hierarchy will be defined and these flows constitute an -folds direct
product of the positive half of the quantum torus symmetries.Comment: 11 pages, to appear in Journal of Geometry and Physic
Sato theory on the -Toda hierarchy and its extension
In this paper, we construct the Sato theory including the Hirota bilinear
equations and tau function of a new -deformed Toda hierarchy(QTH). Meanwhile
the Block type additional symmetry and bi-Hamiltonian structure of this
hierarchy are given. From Hamiltonian tau symmetry, we give another definition
of tau function of this hierarchy. Afterwards, we extend the -Toda hierarchy
to an extended -Toda hierarchy(EQTH) which satisfy a generalized Hirota
quadratic equation in terms of generalized vertex operators. The Hirota
quadratic equation might have further application in Gromov-Witten theory. The
corresponding Sato theory including multi-fold Darboux transformations of this
extended hierarchy is also constructed. At last, we construct the
multicomponent extension of the -Toda hierarchy and show the integrability
including its bi-Hamiltonian structure, tau symmetry and conserved densities.Comment: 28 Pages. arXiv admin note: substantial text overlap with
arXiv:1403.068
Extensions of the finite nonperiodic Toda lattices with indefinite metrics
In this paper, we firstly construct a weakly coupled Toda lattices with
indefinite metrics which consist of different coupled Hamiltonian systems.
Afterwards, we consider the iso-spectral manifolds of extended tridiagonal
Hessenberg matrix with indefinite metrics what is an extension of a strict
tridiagonal matrix with indefinite metrics. For the initial value problem of
the extended symmetric Toda hierarchy with indefinite metrics, we introduce the
inverse scattering procedure in terms of eigenvalues by using the Kodama's
method. In this article, according to the orthogonalization procedure of
Szeg\"{o}, the relationship between the -function and the given Lax
matrix is also discussed. We can verify the results derived from the
orthogonalization procedure with a simple example. After that, we construct a
strongly coupled Toda lattices with indefinite metrics and derive its tau
structures. At last, we generalize the weakly coupled Toda lattices with
indefinite metrics to the -Toda lattices with indefinite metrics.Comment: 18 Pages, to appear in Journal of Statistical Physic
Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation
In this paper, we derive Darboux transformation of the inhomogeneous Hirota
and the Maxwell-Bloch(IH-MB) equations which is governed by femtosecond pulse
propagation through inhomogeneous doped fibre. The determinant representation
of Darboux transformation is used to derive soliton solutions, positon
solutions of the IH-MB equations.Comment: accepted by SCIENCE CHINA Physics, Mechanics & Astronomy. arXiv admin
note: substantial text overlap with arXiv:1205.119
Rogue waves of the Frobenius nonlinear Schr\"odinger equation
In this paper, by considering the potential application in two mode nonlinear
waves in nonlinear fibers under a certain case, we define a coupled nonlinear
Schr\"odinger equation(called Frobenius NLS equation) including its Lax pair.
Afterwards, we construct the Darboux transformations of the Frobenius NLS
equation. New solutions can be obtained from known seed solutions by the
Dardoux transformations. Soliton solutions are generated from trivial seed
solutions. Also we derive breather solutions of the Frobenius NLS
equation obtained from periodic seed solutions. Interesting enough, we find the
amplitudes of vary in size in different areas with period-like fluctuations
in the background. This is very different from the solution of the
single-component classical nonlinear Schr\"odinger equation. Then, the rogue
waves of the Frobenius NLS equation are given explicitly by a Taylor series
expansion about the breather solutions . Also the graph of rogue wave
solution shows that the rogue wave has fluctuations around the peak. The
reason of this phenomenon should be in the dynamical dependence of on
which is independent on .Comment: 14 Page
Additional symmetry of the modified extended Toda hierarchy
In this paper, one new integrable modified extended Toda hierarchy(METH) is
constructed with the help of two logarithmic Lax operators. With this
modification, the interpolated spatial flow is added to make all flows
complete. To show more integrable properties of the METH, the bi-Hamiltonian
structure and tau symmetry of the METH will be given. The additional symmetry
flows of this new hierarchy are presented. These flows form an infinite
dimensional Lie algebra of Block type.Comment: 13 page
Quantum Torus symmetry of the KP, KdV and BKP hierarchies
In this paper, we construct the quantum Torus symmetry of the KP hierarchy
and further derive the quantum torus constraint on the tau function of the KP
hierarchy. That means we give a nice representation of the quantum Torus Lie
algebra in the KP system by acting on its tau function. Comparing to the
symmetry, this quantum Torus symmetry has a nice algebraic
structure with double indices. Further by reduction, we also construct the
quantum Torus symmetries of the KdV and BKP hierarchies and further derive the
quantum Torus constraints on their tau functions. These quantum Torus
constraints might have applications in the quantum field theory, supersymmetric
gauge theory and so on.Comment: published in Lett. Math. Phys. online ahead of print 15 August 201
Dispersionless bigraded Toda Hierarchy and its additional symmetry
In this paper, we firstly give the definition of dipersionless bigraded Toda
hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define
Orlov-Schulman's \M_L, \M_R operator and give the additional Block symmetry
of dBTH. Meanwhile we give tau function of dBTH and some some related
dipersionless bilinear equations.Comment: 31 pages, Accepted by Reviews in Mathematical Physic
Multi-fold Darboux transformations of the extended bigraded Toda hierarchy
With the extended logarithmic flow equations and some extended Vertex
operators in generalized Hirota bilinear equations, extended bigraded Toda
hierarchy(EBTH) was proved to govern the Gromov-Witten theory of orbiford
in literature. The generating function of these Gromov-Witten
invariants is one special solution of the EBTH. In this paper, the multi-fold
Darboux transformations and their determinant representations of the EBTH are
given with two different gauge transformation operators. The two Darboux
transformations in different directions are used to generate new solutions from
known solutions which include soliton solutions of -EBTH, i.e. the EBTH
when . From the generation of new solutions, one can find the big
difference between the EBTH and the extended Toda hierarchy(ETH). Meanwhile we
plotted the soliton graphs of the -EBTH from which some approximation
analysis will be given. From the analysis on velocities of soliton solutions,
the difference between the extended flows and other flows are shown. The two
different Darboux transformations constructed by us might be useful in
Gromov-Witten theory of orbiford .Comment: 26 pages, accepted by Zeitschrift f\"ur Naturforschung
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