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Register a new userKadomtsev-Petviashvili turning points and CKP hierarchy
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A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. The new identity for theta-functions of curves with holomorphic involution having fixed points is obtained.
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This is a short review of the Kadomtsev-Petviashvili hierarchies of types B and C. The main objects are the $L$-operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function and the tau-function. All of them are discussed in the paper. The connections with the usual Kadomtsev-Petviashvili hierarchy (of the type A) are clarified. Examples of soliton solutions and the dispersionless limit of the hierarchies are also considered.
We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $bar {cal L}={cal L}^{dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.
A generalized Kadomtsev-Petviashvili equation, describing water waves in oceans of varying depth, density and vorticity is discussed. A priori, it involves 9 arbitrary functions of one, or two variables. The conditions are determined under which the equation allows an infinite dimensional symmetry algebra. This algebra can involve up to three arbitrary functions of time. It depends on precisely three such functions if and only if it is completely integrable.
The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey-Stewartson equation cite{FDS2009}, in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d-bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial-value problems in 2+1 is that the d-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.
A new method for the solution of initial-boundary value problems for textit{linear} and textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 cite{F1997}. This approach was subsequently extended to initial-boundary value problems for evolution PDEs in two spatial dimensions, first in the case of linear PDEs cite{F2002b} and, more recently, in the case of integrable nonlinear PDEs, for the Davey-Stewartson and the Kadomtsev-Petviashvili II equations on the half-plane (see cite{FDS2009} and cite{MF2011} respectively). In this work, we study the analogous problem for the Kadomtsev-Petviashvili I equation; in particular, through the simultaneous spectral analysis of the associated Lax pair via a d-bar formalism, we are able to obtain an integral representation for the solution, which involves certain transforms of all the initial and the boundary values, as well as an identity, the so-called global relation, which relates these transforms in appropriate regions of the complex spectral plane.
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