Tau function (integrable systems)
Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains.
They were originally introduced by Ryogo Hirota[1] in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.
-function, was first used systematically by Mikio Sato[2] and his students[3][4] in the specific context of the Kadomtsev–Petviashvili (or KP) equation and related integrable hierarchies.
-function satisfying a Hirota-type system of bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), the corresponding solutions of the equations of the integrable hierarchy are explicitly expressible in terms of it and its logarithmic derivatives up to a finite order.
-function may either be: 1) an analytic function of a finite or infinite number of independent, commuting flow variables, or deformation parameters; 2) a discrete function of a finite or infinite number of denumerable variables; 3) a formal power series expansion in a finite or infinite number of expansion variables, which need have no convergence domain, but serves as generating function for certain enumerative invariants appearing as the coefficients of the series; 4) a finite or infinite (Fredholm) determinant whose entries are either specific polynomial or quasi-polynomial functions, or parametric integrals, and their derivatives; 5) the Pfaffian of a skew symmetric matrix (either finite or infinite dimensional) with entries similarly of polynomial or quasi-polynomial type.
In the Hamilton–Jacobi approach to Liouville integrable Hamiltonian systems, Hamilton's principal function, evaluated on the level surfaces of a complete set of Poisson commuting invariants, plays a role similar to the
-function, serving both as a generating function for the canonical transformation to linearizing canonical coordinates and, when evaluated on simultaneous level sets of a complete set of Poisson commuting invariants, as a complete solution of the Hamilton–Jacobi equation.
-function of isospectral type is defined as a solution of the Hirota bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), from which the linear operator undergoing isospectral evolution can be uniquely reconstructed.
Geometrically, in the Sato[2] and Segal-Wilson[11] sense, it is the value of the determinant of a Fredholm integral operator, interpreted as the orthogonal projection of an element of a suitably defined (infinite dimensional) Grassmann manifold onto the origin, as that element evolves under the linear exponential action of a maximal abelian subgroup of the general linear group.
It typically arises as a partition function, in the sense of statistical mechanics, many-body quantum mechanics or quantum field theory, as the underlying measure undergoes a linear exponential deformation.
For the more general case of linear ordinary differential equations with rational coefficients, including irregular singularities, they are developed in reference.
This equation plays a prominent role in plasma physics and in shallow water ocean waves.
gives an infinite sequence of functions that satisfy further systems of nonlinear autonomous PDE's, each involving partial derivatives of finite order with respect to a finite number of the KP flow parameters
-function provides a solution, at least in the formal sense, of this infinite system of nonlinear partial differential equations.
functions the Schlesinger equations (8) imply that the differential form on the space of parameters is closed: and hence, locally exact.
of the matrices appearing in (6), which is an invariant of the Schlesinger equations, equal to a constant, and quotienting by its stabilizer under
[15][16][17] For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes matrices and connection matrices, and there are further isomonodromic deformation parameters associated with the local asymptotics, but the isomonodromic
[12] There is similarly a Poisson bracket structure on the space of rational matrix valued functions of the spectral parameter
and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics.
via exterior and interior multiplication by the basis elements and satisfy the canonical anti-commutation relations These generate the standard fermionic representation of the Clifford algebra on the direct sum
This is annihilated by the following operators The dual fermionic Fock space vacuum state, denoted
positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition
pairs of creation and annihilation operators, labelled by the Frobenius indices The integers
indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while
of the infinite dimensional Grassmannian consisting of the orbit, under the action of the general linear group
of the fermionic Fock space, which are equivalent to the Hirota bilinear residue relation (1).
can be expressed as the fermionic vacuum state expectation value (VEV): where is the abelian subgroup of
themselves, which correspond to the special element of the Grassmann manifold whose image under the Plücker map is
be a conjugation invariant integrable density function Define a deformation family of measures for small
of the infinite sequence of auxiliary variables defined by: the function is a double KP
