Two-component KP Research Articles

Gauge transformations of two-component KP and mKP hierarchies

Based on generalized Lax equations and the Sato theory on KP and mKP hierarchies derived from the basic algebra of pseudo-differential operators, Miura- and auto-Bäcklund transformations can be derived from the gauge transformations of the Lax operators. In this paper, we study the gauge transformations of two-component KP and mKP hierarchies using the two-component generalized gauge transformations of the original operators.

A class of reductions of the two-component KP hierarchy and the Hirota–Ohta system

We introduce a class of reductions of the two-component KP hierarchy, which includes the Hirota-Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation characterising the reduction. We derive the reduction conditions in terms of the Lax operator and higher linear operators of the hierarchy, as well as in terms of the basic two-component KP system of equations.

A square root of Hurwitz numbers

We exhibit a generating function of spin Hurwitz numbers analogous to (disconnected) double Hurwitz numbers that is a tau function of the two-component BKP (2-BKP) hierarchy and is a square root of a tau function of the two-component KP (2-KP) hierarchy defined by related Hurwitz numbers.

Discrete local induction equation

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger equation. In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation. We also present explicit formulas for both smooth and discrete curves in terms of $\tau$ functions of the two-component KP hierarchy.

General soliton solutions to a $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

We investigate a $$(2+1)$$ -dimensional nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. By employing the Hirota’s bilinear method and the KP hierarchy reduction method, general soliton solutions to the $$(2+1)$$ -dimensional nonlocal NLS equation with zero and nonzero boundary conditions are derived. These solutions are given in forms of Gram-type determinants. We first construct general bright solitons with zero boundary condition by constraining the tau functions of two-component KP hierarchy. Furthermore, we derive general dark and antidark solitons with nonzero boundary from the tau functions of single-component KP hierarchy.

Recursion relations for the constrained multi-component KP hierarchy

In this paper, we define a new constrained multi-component KP(cMKP) hierarchy which contains the constrained KP(cKP) hierarchy as a special case. We derive the recursion operator of the constrained multi-component KP hierarchy. As a special example, we also give the recursion operator of the constrained two-component KP hierarchy.

Pfaffian and Determinantal Tau Functions

Adler, Shiota and van Moerbeke observed that a tau function of the Pfaff lattice is a square root of a tau function of the Toda lattice hierarchy of Ueno and Takasaki. In this paper we give a representation theoretical explanation for this phenomenon. We consider 2-BKP and two-component 2-KP tau functions. We shall show that a square of a BKP tau function is equal to a certain two-component KP tau function and a square of a 2-BKP tau function is equal to a certain two-component 2-KP tau function.

Two-component soliton systems and the Painlevé equations

We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear 2 × 2 system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the Schlesinger system and the sixth Painlevé equation is given from this linear system. A unified approach to treat the other Painlevé equations from the usual two-component KP hierarchy is also considered.

Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions

A new representation of the 2N fold integrals appearing in various two-matrix models that admit reductions to integrals over their eigenvalues is given in terms of vacuum state expectation values of operator products formed from two-component free fermions. This is used to derive the perturbation series for these integrals under deformations induced by exponential weight factors in the measure, expressed as double and quadruple Schur function expansions, generalizing results obtained earlier for certain two-matrix models. Links with the coupled two-component KP hierarchy and the two-component Toda lattice hierarchy are also derived.

The extended supersymmetrization of the multicomponent Kadomtsev - Petviashvilli hierarchy

We describe three different approaches to the extended (N = 2) supersymmetrization of the multicomponent KP hierarchy. In the first one we utilize only superfermions while in the second only superbosons and in the third superbosons as well as superfermions. It is shown that many soliton equations can be embedded in the supersymmetry theory by using the first approach even if we do not change these equations in the bosonic limit of the supersymmetry. In the second or third approach we obtain a generalization of the soliton equations in the bosonic limit which remains in the class of the usual commuting functions. As a byproduct of our analysis we prove that for the first procedure the bosonic part of the one-component supersymmetric KP hierarchy coincides with the usual classical two-component KP hierarchy.

Two-component KP hierarchy and the classical Boussinesq equation

It has been shown that solutions to the KP equation and other equations in the KP hierarchy take a Wronskian form. The authors extend these ideas to two-component Wronskians and the Hirota equations (1986) which they satisfy. By a reduction procedure they show how dependence on some of the variables can be removed and look in detail at the classical Boussinesq equation and its modifications and show how these equations fit into the two-component KP hierarchy.

Conserved quantities of two-component KP hierarchy

The two-component KP hierarchy is considered. It is shown that the Davey-Stewartson equation is the simplest nontrivial equation in the hierarchy, and that the nonlinear Schrödinger equation, its higher order extension and the modified KdV equation are derived from suitable reductions of the hierarchy. By using this fact, conserved quantities of those equations are explicitly given.

On the A 1(1) reduction of the two-component KP hierarchy

The reduction of the two-component KP hierarchy with respect to the affine Lie algebra A 1 (1) is found to be equivalent to the hierarchy of integrable systems associated with an energy-dependent Schrödinger operator. The classical Boussinesq system turns out to be the reduction of a recently introduced (2 + 1)-dimensional integrable model.