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The Cauchy matrix approach is developed for solving nonisospectral Kadomtsev–Petviashvili equation and the nonisospectral modified Kadomtsev–Petviashvili equation. By means of a Sylvester equation , a set of scalar master functions are defined. We derive the evolution of scalar functions using the nonisospectral dispersion relations. Some explicit solutions are illustrated together with the analysis of their dynamics. [ABSTRACT FROM AUTHOR]

We construct the parity–time symmetric solitons of the complex KP equation using the totally nonnegative Grassmannian. We obtain that every element in the totally nonnegative orthogonal Grassmannian corresponds to a parity–time symmetric soliton solution. [ABSTRACT FROM AUTHOR]

We develop a direct method, the Cauchy matrix approach, to construct matrix solutions of noncommutative soliton equations. This approach is based on the Sylvester equation, and solutions can be presented without using quasideterminants. The matrix Kadomtsev–Petviashvili equation with self-consistent sources is employed as an example to demonstrate the approach. As a reduction, explicit solutions of the matrix Mel'nikov model for long–short wave interaction are obtained. [ABSTRACT FROM AUTHOR]

We study the system of discrete equations on the quadrilateral graph. We introduce the notion of the set of independent minimal-order integrals along the characteristic directions, as well as the concept of the characteristic Lie–Rinehart algebra for the system of equations on the graph. We prove that the system admits the complete set of integrals along the considered direction if and only if the dimension of the characteristic algebra corresponding to this direction is finite. In other words, the system is Darboux-integrable if and only if its characteristic algebras in both directions are finite dimensional. As examples of Darboux-integrable systems of discrete equations on quadrilateral graphs we consider reductions of Hirota–Miwa equation, the -system, and the Kadomtsev–Petviashvili lattice equation and construct the characteristic algebras for them. [ABSTRACT FROM AUTHOR]

We introduce a class of reductions of the two-component KP hierarchy that includes the Hirota–Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation characterizing 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. [ABSTRACT FROM AUTHOR]

We study elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation and derive the equations of motion of their poles. The auxiliary linear problems for the wave function are the main technical tool. [ABSTRACT FROM AUTHOR]

Nonlinear generalizations of integrable equations in one dimension, such as the Korteweg-de Vries and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting from the analytic standpoint because of their critical behavior. We study analogous nonlinear p-power generalizations of the integrable Kadomtsev-Petviashvili and Boussinesq equations in two dimensions. For all p ≠ 0, we present a Hamiltonian formulation of these two generalized equations. We derive all Lie symmetries including those that exist for special powers p ≠ 0. We use Noether’s theorem to obtain conservation laws arising from the variational Lie symmetries. Finally, we obtain explicit line soliton solutions for all powers p > 0 and discuss some of their properties. [ABSTRACT FROM AUTHOR]

We construct several types of Darboux transformations for the discrete Kadomtsev-Petviashvili equation with self-consistent sources (dKPwS) including the elementary Darboux transformation, the adjoint Darboux transformation, and the binary Darboux transformation. These Darboux transformations can be used to obtain some solutions of the dKPwS. We give some solutions explicitly. [ABSTRACT FROM AUTHOR]

We construct solutions of the Kadomtsev-Petviashvili-I equation in terms of Fredholm determinants. We deduce solutions written as a quotient of Wronskians of order 2N. These solutions, called solutions of order N, depend on 2N−1 parameters. They can also be written as a quotient of two polynomials of degree 2N(N +1) in x, y, and t depending on 2N−2 parameters. The maximum of the modulus of these solutions at order N is equal to 2(2N + 1)2. We explicitly construct the expressions up to the order six and study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters. [ABSTRACT FROM AUTHOR]

In the tropical limit of matrix KP-II solitons, their support at a fixed time is a planar graph with “polarizations” attached to its linear parts. We explore a subclass of soliton solutions whose tropical limit graph has the form of a rooted and generically binary tree and also solutions whose limit graph comprises two relatively inverted such rooted tree graphs. The distribution of polarizations over the lines constituting the graph is fully determined by a parameter-dependent binary operation and a Yang-Baxter map (generally nonlinear), which becomes linear in the vector KP case and is hence given by an R-matrix. The parameter dependence of the binary operation leads to a solution of the pentagon equation, which has a certain relation to the Rogers dilogarithm via a solution of the hexagon equation, the next member in the family of polygon equations. A generalization of the R-matrix obtained in the vector KP case also solves a pentagon equation. A corresponding local version of the latter then leads to a new solution of the hexagon equation. [ABSTRACT FROM AUTHOR]

We construct lump solutions of the Kadomtsev-Petviashvili-I equation using Grammian determinants in the spirit of the works by Ohta and Yang. We show that the peak locations depend on the real roots of the Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. We also prove that if the time goes to −∞, then all the peak locations are on a vertical line, while if the time goes to ∞, then they are all on a horizontal line, i.e., a π/2 rotation is observed after interaction. [ABSTRACT FROM AUTHOR]

We study a new property of integrable systems, the existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional hydrodynamic chains. We describe an effective algorithm for successively computing an infinite set of three-dimensional conservation laws for the Benney pair of commuting flows. [ABSTRACT FROM AUTHOR]

We propose a quantization of the Kadomtsev-Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms $$\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}$$ and $$\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}$$ for all combinations of particles with masses m , m , and m + m for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We construct Virasoro-type additional symmetries of a kind of constrained multicomponent Kadomtsev-Petviashvili (KP) hierarchy and obtain the Virasoro flow equation for the eigenfunctions and adjoint eigenfunctions. We show that the algebraic structure of the Virasoro symmetry is retained under discretization from the constrained multicomponent KP hierarchy to the discrete constrained multicomponent KP hierarchy. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We describe the properties of the Cauchy-Jost (also known as Cauchy-Baker-Akhiezer) function of the Kadomtsev-Petviashvili-II equation. Using the $$\bar \partial $$-method, we show that for this function, all equations of the Kadomtsev-Petviashvili-II hierarchy are given in a compact and explicit form, including equations for the Cauchy-Jost function itself, time evolutions of the Jost solutions, and evolutions of the potential of the heat equation. [ABSTRACT FROM AUTHOR]

We consider cohomological and Poisson structures associated with the special tautological subbundles $$TB_{W_{1,2, \ldots n} }$$ for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of $$TB_{W_{1,2, \ldots n} }$$ are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev-Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in $$TB_{W_{1,2, \ldots n} }$$ can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in $$TB_{W_{\hat S} }$$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures. [ABSTRACT FROM AUTHOR]

We split the algebra of pseudodifferential operators in two different ways into the direct sum of two Lie subalgebras and deform the set of commuting elements in one subalgebra in the direction of the other component. The evolution of these deformed elements leads to two compatible systems of Lax equations that both have a minimal realization. We show that this Lax form is equivalent to a set of zero-curvature relations. We conclude by presenting linearizations of these systems, which form the key framework for constructing the solutions. [ABSTRACT FROM AUTHOR]

We consider the heat operator with a general multisoliton potential and derive its extended resolvent depending on a parameter q ∈ ℝ. We show that it is bounded in all variables and find its singularities in q. We introduce the Green's functions and study their properties in detail. [ABSTRACT FROM AUTHOR]

We discuss some interesting aspects of the wave breaking in localized solutions of the dispersionless Kadomtsev-Petviashvili equation, an integrable partial differential equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in 2+1 dimensions, which arise in several physical contexts such as acoustics, plasma physics, and hydrodynamics. For this, we use an inverse spectral transform for multidimensional vector fields that we recently developed and, in particular, the associated inverse problem, a nonlinear Riemann-Hilbert problem on the real axis. In particular, we discuss how the derivative of the solution blows up at the first breaking point in any direction of the plane (x, y) except in the transverse breaking direction and how the solution becomes three-valued in a compact region of the plane (x, y) after the wave breaking. [ABSTRACT FROM AUTHOR]

We consider the question of integrable boundary-value problems in the examples of the two-dimensional Toda chain and Kadomtsev-Petviashvili equation. We discuss the problems that are integrable from the standpoints of two basic definitions of integrability. As a result, we propose a method for constructing a hierarchy of integrable boundary-value problems where the boundaries are cylindric surfaces in the space of three variables. We write explicit formulas describing wide classes of solutions of these boundary-value problems for the two-dimensional Toda chain and Kadomtsev-Petviashvili equation. [ABSTRACT FROM AUTHOR]

We study properties of the purely solitonic τ-function and potential of the heat equation in detail. We describe the asymptotic behavior of the potential and establish the ray structure of this asymptotic behavior on the plane (x, x) in dependence on the parameters of the potential. [ABSTRACT FROM AUTHOR]

We consider the nonstationary Schrodinger equation with the potential being a perturbation of a generic one-dimensional potential by means of a decaying two-dimensional function in the framework of the extended resolvent approach. We give the corresponding modification of the Jost and advanced/retarded solutions and spectral data and present relations between them. [ABSTRACT FROM AUTHOR]

We propose a method for solving the (2+1)-dimensional Kadomtsev-Petviashvili equation with negative dispersion (KP-II) using the second and third members of the disipative version of the AKNS hierarchy. We show that dissipative solitons (dissipatons) of those members yield the planar solitons of the KP-II. From the Hirota bilinear form of the SL(2, ℝ) AKNS flows, we formulate a new bilinear representation for the KP-II, by which we construct one- and two-soliton solutions and study the resonance character of their mutual interactions. Using our bilinear form, for the first time, we create a four-virtual-soliton resonance solution of the KP-II, and we show that it can be obtained as a reduction of a four-soliton solution in the Hirota-Satsuma bilinear form for the KP-II. [ABSTRACT FROM AUTHOR]

We study the boundary value problem for the Kadomtsev-Petviashvili equation on the half-plane y>0 with a homogeneous condition along the boundary. We show that the problem can be efficiently solved using the dressing method. We present explicit solutions for particular cases of the boundary value problem. [ABSTRACT FROM AUTHOR]