7,726 research outputs found
Dispersionless Hierarchies, Hamilton-Jacobi Theory and Twistor Correspondences
The dispersionless KP and Toda hierarchies possess an underlying twistorial
structure. A twistorial approach is partly implemented by the method of
Riemann-Hilbert problem. This is however still short of clarifying geometric
ingredients of twistor theory, such as twistor lines and twistor surfaces. A
more geometric approach can be developed in a Hamilton-Jacobi formalism of
Gibbons and Kodama. AMS Subject Classifiation (1991): 35Q20, 58F07,70H99Comment: 20 pages, latex, no figure
Old and New Reductions of Dispersionless Toda Hierarchy
This paper is focused on geometric aspects of two particular types of
finite-variable reductions in the dispersionless Toda hierarchy. The reductions
are formulated in terms of "Landau-Ginzburg potentials" that play the role of
reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's
trigonometric polynomial. The other is a transcendental function, the logarithm
of which resembles the waterbag models of the dispersionless KP hierarchy. They
both satisfy a radial version of the L\"owner equations. Consistency of these
L\"owner equations yields a radial version of the Gibbons-Tsarev equations.
These equations are used to formulate hodograph solutions of the reduced
hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in
the language of classical differential geometry (Darboux equations, Egorov
metrics and Combescure transformations). Flat coordinates of the underlying
Egorov metrics are presented
Modified melting crystal model and Ablowitz-Ladik hierarchy
This paper addresses the issue of integrable structure in a modified melting
crystal model of topological string theory on the resolved conifold. The
partition function can be expressed as the vacuum expectation value of an
operator on the Fock space of 2D complex free fermion fields. The quantum torus
algebra of fermion bilinears behind this expression is shown to have an
extended set of "shift symmetries". They are used to prove that the partition
function (deformed by external potentials) is essentially a tau function of the
2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be
characterized by a factorization problem of \ZZ\times\ZZ matrices as well.
The associated Lax operators turn out to be quotients of first order difference
operators. This implies that the solution of the 2D Toda hierarchy in question
is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda)
hierarchy. As a byproduct, the shift symmetries are shown to be related to
matrix-valued quantum dilogarithmic functions.Comment: latex2e, 33 pages, no figure; (v2) accepted for publicatio
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