Two-dimensional Toda Lattice Hierarchy Research Articles

Solitons and normal random matrices

We discuss a general relation between the solitons and statistical mechanics and show that the partition function of the normal random matrix model can be obtained from the multi-soliton solutions of the two-dimensional Toda lattice hierarchy in a special limit.

General rogue wave solution to the discrete nonlinear Schrödinger equation

In the present paper, we are concerned with the rogue wave solution to an integrable discrete nonlinear Schrödinger (NLS) equation. First, through the KP–Toda reduction method, the general breather solution of the discrete NLS equation is deduced from two bilinear equations in discrete two-dimensional Toda-lattice hierarchy. Then, by taking an ingenious and successive limit to the breather solution, we derive the higher order rogue wave solution to the discrete NLS equation.

ABJM matrix model and 2D Toda lattice hierarchy

It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions.

Rational reductions of the 2D-Toda hierarchy and mirror symmetry

We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1+1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz–Ladik hierarchy and E. Frenkel's q -deformed Gelfand–Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki–Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the Ablowitz–Ladik hierarchy to Gromov–Witten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi–Yau threefolds obtained from minimal resolutions of the local orbifold line.

Multi-soliton solution to the two-component Hunter–Saxton equation

In this paper, we study the bilinear form and the general N-soliton solution for a two-component Hunter–Saxton (2-HS) equation, which is the short wave limit of a two-component Camassa–Holm equation. By defining a hodograph transformation based on a conservation law and appropriate dependent variable transformations, we propose a set of bilinear equations which yields the 2-HS equation. Furthermore, we construct the N-soliton solution to the 2-HS equation based on the tau functions of an extended two-dimensional Toda-lattice hierarchy through reductions. One- and two-soliton solutions are calculated and analyzed.

Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

We consider two-component integrable generalizations of the dispersionless two-dimensional Toda lattice (2DTL) hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov–Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph-type transformations. Generating equations and Lax–Sato equations are introduced, and a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, and its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.

An extended two-dimensional Toda lattice hierarchy and two-dimensional Toda lattice with self-consistent sources

We extend the two-dimensional Toda lattice hierarchy (2DTLH) by its squared eigenfunction symmetries. This extended 2DTLH (ex2DTLH) includes the two-dimensional Toda lattice equation with self-consistent sources (2DTLSCS) as its first nontrivial equation. Lax representation of ex2DTLH is also presented. With the help of the Lax representation, we construct a nonauto-Bäcklund Darboux transformation (DT) for 2DTLSCS by applying the variation of constants to 2DTLSCS auto-Bäcklund DT. This DT enables us to find many solutions to 2DTLSCS, including solitons, rational solutions, positons, negatons, and complexitons.

Hamiltonian Structures of Fermionic Two-Dimensional Toda Lattice Hierarchies

By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows one to describe these hierarchies in the framework of the Hamiltonian formalism and construct their first two Hamiltonian structures. The first Hamiltonian structure is obtained for both bosonic and fermionic Lax operators while the second Hamiltonian structure is established for bosonic Lax operators only.

Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures

By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N = ( 2 | 2 ) and N = ( 0 | 2 ) supersymmetric TL hierarchies as particular cases. Performing their reduction to the one-dimensional case by imposing suitable constraints we derive the corresponding 1D fermionic TL hierarchies. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with an involution generalizing the graded commutator in superalgebras, which allows one to describe these hierarchies in the framework of the Hamiltonian formalism and construct their first two Hamiltonian structures. The first Hamiltonian structure is obtained for both bosonic and fermionic Lax operators while the second Hamiltonian structure is established for bosonic Lax operators only. We propose the graded modified Yang–Baxter equation in the operator form and demonstrate that for the class of graded antisymmetric R-matrices it is equivalent to the tensor form of the graded classical Yang–Baxter equation.

Semiclassical $$\bar \partial $$ -Method: Generating Equations for Dispersionless Integrable Hierarchies

We use the semiclassical \(\bar \partial \)-dressing method to derive compact generating equations for dispersionless hierarchies. The considered illustrative examples are the dispersionless Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies.

Conservation Laws of the Two-Dimensional Toda Lattice Hierarchy

A novel method of constructing the conservation laws of (1+2)-dimensional differential-difference systems is proposed. By introducing the generalized Riccati equation related to the pseudo-difference operator, we obtain the infinitely many conserved densities and the associated fluxes of the two-dimensional Toda lattice hierarchy. Moreover, this method presents more forms of the conservation laws than the previous approach [K. Kajiwara and J. Satsuma: J. Math. Phys. 32 (1991) 506].

Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂̄-dressing method

The quasiclassical limit of the scalar nonlocal ∂̄-problem is derived and a quasiclassical version of the ∂̄-dressing method is presented. Dispersionless Kadomtsev–Petviashvili (KP), modified KP, and dispersionless two-dimensional Toda lattice (2DTL) hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy is nothing but the ring of symmetries for the quasiclassical ∂̄-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.

On a Discrete Analogue of the Two-Dimensional Toda Lattice Hierarchy

A discrete analogue of the two-dimensional Toda lattice hierarchy, which includes the discrete analogue of the two-dimensional Toda lattice equation, is obtained by extending the theory of the Toda lattice hierarchy. Special solutions and bilinear equations for the discrete analogue of the two-dimensional Toda lattice hierarchy are presented. It is shown that the subjects originated from the Sato theory are useful for the discrete-time integrable systems.

Multivariate hypergeometric functions as τ-functions of Toda lattice and Kadomtsev–Petviashvili equation

We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as τ-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times of those hierarchies. The discrete Toda lattice variable shifts parameters of hypergeometric functions. The role of additional symmetries in generating hypergeometric τ-functions is explained.

Tau-functions generating the conservation laws for generalized integrable hierarchies of KdV and affine toda type

For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This tau-function acts as a partition function for the conserved densities, which fits its potential interpretation as the effective action of some quantum system. The class consists of multi-component generalizations of the Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The relationship between the former and the approach of Feigin, Frenkel and Enriquez to soliton equations of KdV and mKdV type is also discussed. These results considerably simplify the calculation of the conserved charges carried by the soliton solutions to the equations of the hierarchy, which is important to establish their interpretation as particles. By way of illustration, we calculate the charges carried by a set of constrained KP solitons recently constructed.

An algebraic description of generalizedk-constraints

The generalized k-constrained KP hierarchy is shown to correspond to a so-called pseudo-reduction of the two-dimensional Toda lattice hierarchy, described in a free-fermion approach which is adapted to the case of two singularities in the spectral parameter range. Wronskian solutions are discussed and, in particular, soliton solutions are recovered through a reduction of the Toda solitons.

Five-Dimensional Gauge Theories and Whitham–Toda Equation

The five-dimensional supersymmetric SU (N) gauge theory is studied in the framework of the relativistic Toda chain. This equation can be embedded in two-dimensional Toda lattice hierarchy. This system has the conjugate structure which corresponds to the charge conjugation.

Faces of Relativistic Toda Chain

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of the Lax representation, orthogonal polynomial systems, and τ-function approach. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. Some applications to biorthogonal polynomial systems are discussed.

New reductions of the Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies via symmetry constraints

New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy and the two-dimensional Toda lattice (2DTL) hierarchy are considered on the basis of Sato’s approach. Within this approach these hierarchies are represented by infinite sets of equations for potentials u1,u2,u3,..., of pseudodifferential operators and their eigenfunctions ψ and adjoint eigenfunctions ψ*. The KP and the 2DTL hierarchies are studied under constraints of the following type: ∑n=1N αnSn(u1,u2,u3,...)=Ωx, where Sn are symmetries for these hierarchies, αn are arbitrary constants, and Ω is an arbitrary linear functional of the quantity ψ(λ)ψ*(μ). It is shown that for the KP hierarchy these constraints give rise to hierarchies of 1+1-dimensional commuting flows for the variables u2,u3,...,uN,ψ,ψ*. Many known systems and several new ones are among them. Symmetry reductions for the 2DTL hierarchy give rise both to finite-dimensional dynamical systems and 1+1-dimensional discrete systems. Some few results for the modified KP hierarchy are also presented.