mKP Hierarchy Research Articles

On the constrained discrete mKP hierarchies: Gauge transformations and the generalized Wronskian solutions

On the constrained discrete mKP hierarchies: Gauge transformations and the generalized Wronskian solutions

Gauge transformations between three-component KP and three-component mKP hierarchies

Gauge transformations between three-component KP and three-component mKP hierarchies

On the constrained $q$-mKP hierarchy: Additional symmetry and a hidden Virasoro algebraic structure

Построена дополнительная симметрия иерархии $q$-деформированного модифицированного уравнения Кадомцева-Петвиашвили со связями. Введен новый модифицированный оператор. Получены потоки и дополнительные потоки, действующие на модифицированный оператор. Представлены дополнительные потоки, действующие на собственную функцию и на сопряженную собственную функцию. Найдена скрытая структура алгебры Вирасоро в дополнительной симметрии иерархии $q$-деформированного модифицированного уравнения Кадомцева-Петвиашвили со связями.

On the constrained $$q$$-mKP hierarchy: Additional symmetry and a hidden Virasoro algebraic structure

On the constrained $$q$$-mKP hierarchy: Additional symmetry and a hidden Virasoro algebraic structure

Virasoro symmetries of the constrained dispersionless mKP hierarchy

This paper aims at the auto-Bäcklund transformations for the dispersionless mKP hierarchy, dispersionless Miura map between the constrained dispersionless mKP hierarchy and the constrained dispersionless Harry Dym hierarchy and Virasoro symmetries of the constrained dispersionless mKP hierarchy. The additional symmetries of the constrained dispersionless mKP hierarchy are constructed in a manner similar to that of the dispersionless mKP hierarchy. By comparison, a new Laurent series Yl is introduced. The additional flows form a subalgebra of the Virasoro algebra.

Boson–Fermion correspondence of the multi-component constrained mKP hierarchy

We generalize the constrained modified KP (mKP) hierarchy to the multi-component version whose tau function can be expressed as the vacuum expectation value of Clifford operators using free fermions. Meanwhile, we extend the relevant theory of the Boson–Fermion correspondence in single component constrained mKP hierarchy to the multi-component version. This leads us to obtain the rational and soliton solutions of the multi-component constrained mKP hierarchy utilizing the Boson–Fermion correspondence. Furthermore, we give five sets of differential Fay identities and two sets of difference Fay identities of the multi-component mKP hierarchy.

Nonlocal integrable equations from the mKP hierarchy

Nonlocal integrable equations from the mKP hierarchy

The generalized q-Wronskian solutions of the q-deformed constrained modified KP hierarchy

In this paper, we give the form of [Formula: see text]-cmKP hierarchy generated by the gauge transformation operator [Formula: see text]. We show a necessary and sufficient condition to reduce the generalized [Formula: see text]-Wronskian solutions from the [Formula: see text]-mKP hierarchy to generalize [Formula: see text]-Wronskian solutions of [Formula: see text]-component [Formula: see text]-cmKP hierarchy.

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.

Squared eigenfunction symmetries of the constrained integrable hierarchies

In this paper, we mainly investigate the constrained Kadomtsev–Petviashvili (KP) and modified KP (mKP) hierarchies. Squared eigenfunction (SE) symmetry is an important symmetry defined by eigenfunctions and adjoint eigenfunctions, which can be viewed as the generator of the additional symmetries. Due to special constraints on the Lax operators, it is found that there are three different cases for the generating eigenfunctions and adjoint eigenfunctions of the SE symmetries. Then we discuss the changes of the SE symmetries under the Miura links between the KP and mKP hierarchies. Next, the exponential of the SE symmetries is shown to be the binary Darboux transformations. At last based upon the results above, new additional symmetries for the constrained KP and mKP hierarchies are constructed, different from the additional Virasoro symmetries defined by Aratyn.

The reduction of the q-Wronskian solutions of the q-deformed modified KP hierarchy to its constrained case

By using the eigenfunction symmetry constraints of the [Formula: see text]-deformed modified Kadomtsev–Petviashvili ([Formula: see text]-mKP) hierarchy, we show a necessary and sufficient condition to reduce [Formula: see text]-Wronskian solutions of the [Formula: see text]-mKP hierarchy to the [Formula: see text]-cmKP ([Formula: see text]-deformed constrained modified Kadomtsev–Petviashvili) hierarchy defined by setting the Lax operator as [Formula: see text]. Then an illustrative example is given.

On generalized Lax equation of the Lax triple of mKP hierarchy

Based on the Lax pair [Formula: see text] of the mKP hierarchy and the operator Nambu 3-bracket, we propose the generalized Lax equation with respect to the Lax triple [Formula: see text]. For different pairs [Formula: see text] in the generalized Lax equation, a generalized hierarchy including the mKP hierarchy is derived.

Quasilinear systems of Jordan block type and the mKP hierarchy

Hydrodynamic type systems in Riemann invariants arise in a whole range of applications in fluid dynamics, Whitham averaging procedure, differential geometry and the theory of Frobenius manifolds. In this paper we discuss parabolic (Jordan block) analogues of diagonalisable systems. Our main observation is that integrable quasilinear systems of Jordan block type are parametrised by solutions of the modified Kadomtsev–Petviashvili hierarchy. Such systems appear naturally as degenerations of quasilinear systems associated with multi-dimensional hypergeometric functions, in the context of parabolic regularisation of the Riemann equation, as finite-component reductions of hydrodynamic chains, and as hydrodynamic reductions of linearly degenerate dispersionless integrable PDEs in multi-dimensions.

Solutions of the constrained mKP hierarchy by Boson-Fermion correspondence

In this paper, the Hirota bilinear equation of the constrained modified KP hierarchy is expressed as the vacuum expectation values of Clifford operators by using the free fermions method of mKP hierarchy. Then we mainly use the Boson-Fermion correspondence to solve the Hirota bilinear equation of the k-constrained mKP hierarchy. Further, by choosing special group elements in GL∞, the corresponding rational and soliton solutions are given.

Miura and auto-Bäcklund transformations for the discrete KP and mKP hierarchies and their constrained cases

In this paper, the Miura and auto-Bäcklund transformations for the dKP and mdKP hierarchies and their constrained cases are investigated. The combination of the Miura and reverse-Miura transformations are used to derive the auto-Bäcklund transformations. Also some applications of above results are considered.

The q-deformed mKP hierarchy with two parameters

In this paper, with the help of the biparametric quantum calculus we construct the Sato theory on the q-deformation modified Kadomtsev–Petviashvili hierarchy with two parameters (qp-mKP), which is a new deformation of classical mKP hierarchy. The Lax equation and dressing operator of qp-mKP hierarchy are derived. By considering the M operator and [Formula: see text] operator, the additional symmetry of qp-mKP hierarchy is obtained.

The gauge transformation of the modified KP hierarchy

In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators Ti with i = 1, 2, 3 for the mKP hierarchy in Kupershmidt-Kiso version, and find that the gauge transformation operators Ti can not commute with each other. Then two types of gauge transformation operators TD and TI constructed from Ti are proved that they can commute with each other. In particular, TI is introduced for the first time in the literature. And the successive applications of TD and TI in the form of T(n,k), which is the product of n terms of TD and k terms of TI, are derived in three cases for different n and k. At last, the corresponding successive applications of TD and TI on the eigenfunction Φ, the adjoint eigenfunction Ψ and the tau functions τ0 and τ1 are considered.

Lax Triad Approach to Symmetries of Scalar Modified Kadomtsev–Petviashvili Hierarchy**Supported by the National Natural Science Foundation of China under Grant No. 11371241

By means of Lax triads we reconstruct isospectral and nonisospectral scalar modified Kadomtsev–Petviashvili (mKP) hierarchies. In this approach the argument y is treated as an independent variable which is independent of time parameters . Consequently, the isospectral and nonisospectral scalar mKP flows can have clear zero curvature representations, which enables us to handle investigation of symmetries of the scalar isospectral mKP hierarchy as freely as for (1+1)-dimensional systems. As a result, we obtain Lie algebraic structures of the scalar mKP flows and construct symmetries for the scalar isospectral mKP hierarchy.

New extension of dispersionless Harry Dym hierarchy

The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.

Two (2+1)-dimensional expanding dynamical systems associated to the mKP hierarchy

An isospectral problem with a parameter ϵ is presented, for which a modified KdV (mKdV) hierarchy is easily derived from the Tu scheme. Then two different expanding integrable models are obtained, respectively. The main purpose for this focuses on deriving a (2+1)-dimensional mKdV hierarchy (called the mKP hierarchy) and the corresponding different (2+1)-dimensional expanding models (including the linear and nonlinear), whose Hamiltonian structures are obtained by employing an identity developed by us in the paper. As the reduced consequences, the linear and nonlinear (2+1)-dimensional expanding models of the mKP equation are generated.