A new extended matrix KP hierarchy and its solutions

Starting from the matrix KP hierarchy and adding a new τ_B flow, we obtain a new extended matrix KP hierarchy and its Lax representation with the symmetry constraint on squared eigenfunctions taken into account. The new hierarchy contains two sets of times t_A and τ_B and also eigenfunctions and adjoint eigenfunctions as components. We propose a generalized dressing method for solving the extended matrix KP hierarchy and present some solutions. We study the soliton solutions of two types of (2+1)-dimensional AKNS equations with self-consistent sources and two types of Davey-Stewartson equations with selfconsistent sources.     

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

We construct bilinear identities for wave functions of an extended B-type Kadomtsev–Petviashvili (BKP) hierarchy containing two types of (2+1)-dimensional Sawada–Kotera equations with a self-consistent source. Introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the τ -function and bilinear identities for this extended BKP hierarchy. The bilinear identities generate all the Hirota bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, we obtain the Hirota bilinear equations for the two types of (2+1)-dimensional Sawada–Kotera equations in explicit form.     

On the nonisospectral modified Kadomtsev-Peviashvili equation

Two types of Grammian solutions to the nonisospectral modified Kadomtsev-Peviashvili (mKP) equation are derived with the help of the Pfaffian derivative formulae. Starting from the Grammian solutions obtained, we construct the nonisospectral mKP equation with self-consistent sources by applying the source generalization procedure, whose solutions can be given simultaneously.     

On a new generalization of dispersionless Kadomtsev–Petviashvili hierarchy and its reductions

The dispersionless Kadomtsev–Petviashvili hierarchy is generalized by introducing two new time series γ_n and σ_k with two parameters η_n and λ_k. By this hierarchy, we obtain the first type, the second type as well as mixed type of dispersionless Kadomtsev–Petviashvili equation with self‐consistent sources and their related conservation equations. In addition, the reduction and constrained flow of this new hierarchy are studied. The first type, the second type and the mixed type of dispersionless Korteweg–de Vries equation with self‐consistent sources and of dispersionless Boussinesq equation with self‐consistent sources are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the extended dispersionless Toda hierarchy

We derive dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are just a direct consequence of the genus-zero topological recurrence relation for the topological ?P¹ model. Using the dispersionless Hirota equations, we compute the twopoint functions and express the result in terms of Catalan numbers     

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Backlund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Bäcklund transformations.     

Some types of solutions and generalized binary Darboux transformation for the mKP equation with self-consistent sources

In this paper, an mKP equation with self-consistent sources (mKPESCSs) is structured in the framework of the constrained mKP equation. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation and the N-times repeated Darboux transformation with arbitrary functions at time t for the mKPESCSs which offers a non-auto-Bäcklund transformation between two mKPESCSs with different degrees of sources. With the help of these transformations, some new solutions for the mKPESCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found by taking the special initial solution for auxiliary linear problems and the special functions of t-time.     

Symmetries of the DΔmKP hierarchy and their continuum limits

In the recent paper [Stud. App. Math. 147 (2021), 752], squared eigenfunction symmetry constraint of the differential-difference modified Kadomtsev–Petviashvili (DΔmKP) hierarchy converts the DΔmKP system to the relativistic Toda spectral problem and its hierarchy. In this paper, we introduce a new formulation of independent variables in the squared eigenfunction symmetry constraint, under which the DΔmKP system gives rise to the discrete spectral problem and a hierarchy of the differential-difference derivative nonlinear Schrödinger equation of the Chen–Lee–Liu type. In addition, by introducing nonisospectral flows, two sets of symmetries of the DΔmKP hierarchy and their algebraic structure are obtained. We then present a unified continuum limit scheme, by which we achieve the correspondence of the mKP and the DΔmKP hierarchies and their integrable structures.     

Kernel formula approach to the universal Whitham hierarchy

Based on the kernel formula proposed by Carroll and Kodama, we derive the dispersionless Hirota equations of the universal Whitham hierarchy. We also verify the associativity equations in this hierarchy from the dispersionless Hirota equations and give a realization of the associative algebra with the structure constants expressed in terms of residue formulas.     

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two‐component extended Kadomtsew‐Petviashvili (KP) hierarchy. As a by‐product, the N‐soliton solutions in the form of determinants for these equations are provided.     

Dispersionless integrable equations as coisotropic deformations: Extensions and reductions

We discuss the interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain associative algebras and other algebraic structures. We show that with this approach, the dispersionless Hirota equations for the dKP hierarchy are just the associativity conditions in a certain parameterization. We consider several generalizations and demonstrate that B-type dispersionless integrable hierarchies, such as the dBKP and the dVN hierarchies, are coisotropic deformations of the Jordan triple systems. We show that stationary reductions of the dispersionless integrable equations are connected with dynamical systems on the plane that are completely integrable on a fixed energy level. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 439–457, June, 2007.     

Conservation laws for a super G-J hierarchy with self-consistent sources

Based on a well known super Lie algebra, a super integrable system is presented. Then, the super G-J hierarchy with self-consistent sources are obtained. Furthermore, we establish the infinitely many conservation laws for the integrable super G-J hierarchy. The methods derived by us can be generalized to other nonlinear equations hierarchies with self-consistent sources.     

Relation among nonlinear evolution equations and their reductions

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.     

The generalized Kupershmidt deformation for the fifth-order coupled KdV equations hierarchy

Based on the Kupershmidt deformation, we propose the generalized Kupershmidt deformation (GKD) to construct new systems from integrable bi-Hamiltonian system. As applications, the generalized Kupershmidt deformation of the fifth-order coupled KdV equations hierarchy with self-consistent sources and its Lax representation are presented.     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

A non-isospectral integrable couplings of Volterra lattice hierarchy with self-consistent sources

A kind of non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with [Y.F. Zhang, E.G. Fan, Characteristic Numbers of Matrix Lie Algebras, Commun. Theor. Phys (China) 49 (2008) 845] is presented. As an application example, the hierarchy of non-isospectral cubic Volterra lattice hierarchy with self-consistent sources is derived. Furthermore, we construct a non-isospectral integrable couplings of cubic Volterra lattice hierarchy with self-consistent sources by using the loop algebra .     

Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.     

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.     

Solving soliton equations with self-consistent sources by method of variation of parameters

Starting from the solutions of soliton equations and corresponding eigenfunctions obtained by Darboux transformation, we present a new method to solve soliton equations with self-consistent sources (SESCS) based on method of variation of parameters. The KdV equation with self-consistent sources (KdVSCS) is used as a model to illustrate this new method. In addition, we apply this method to construct some new solutions of the derivative nonlinear Schrödinger equation with self-consistent sources (DNLSSCS) such as phase solution, dark soliton solution, bright soliton solution and breather-type solution.     

Quasideterminant solutions of the extended noncommutative Kadomtsev–Petviashvili hierarchy

We construct a nonauto Darboux transformation for the extended noncommutative Kadomtsev–Petviashvili (ncKP) hierarchy and consequently derive its quasi-Wronskian solution. We also obtain the quasi-Wronskian solution of the ncKP equation with self-consistent sources (ncKPESCS) as a by-product. Finally, we use the direct verification method to prove the quasi-Wronskian solution of the ncKPESCS.