Some new methods and results in the theory of (2+1)-dimensional integrable equations

The general resolvent scheme for solving nonlinear integrable evolution equations is formulated. Special attention is paid to the problem of nontrivial dressing and the corresponding transformation of spectral data. The Kadomtsev-Petviashvili equation is considered as the standard example of integrable models in 2+1 dimensions. Properties of the solutionu(t, x, y) of the Kadomtsev-Petviashvili I equation as well as the corresponding Jost solutions and spectral data with given initial datau(0, x, y) belonging to the Schwartz space are presented.Dipartimento di Fisica dell'Università and Sezione, INFN 73100 Lecce, ITALIA. E-mail: boiti@lecce.infn.it and pempi@lecce.infn.it. Steklov Mathematical Institute, Vavilov Str. 42, Moscow 117966, GSP-1, RUSSIA. E-mail: progreb@qft.mian.su. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 185–200, May, 1994.     

Properties of the solitonic potentials of the heat operator

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 (x₁, x₂) in dependence on the parameters of the potential.     

Commutator identities on associative algebras,the non-Abelian Hirota difference equation and its reductions

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.     

Extended resolvent of the heat operator with a multisoliton potential

We consider the heat operator with a general multisoliton potential and derive its extended resolvent depending on a parameter q ?? ?². 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.     

Quantizing the KdV Equation

We consider the quantization procedure for the Gardner–Zakharov–Faddeev and Magri brackets using the fermionic representation for the KdV field. In both cases, the corresponding Hamiltonians are sums of two well-defined operators. Each operator is bilinear and diagonal with respect to either fermion or boson (current) creation/annihilation operators. As a result, the quantization procedure needs no space cutoff and can be performed on the entire axis. In this approach, solitonic states appear in the Hilbert space, and soliton parameters become quantized. We also demonstrate that the dispersionless KdV equation is uniquely and explicitly solvable in the quantum case.     

Commutator identities and integrable hierarchies

Theoretical and Mathematical Physics - The approach based on commutator identities for elements of associative algebras was previously effectively used to investigate $$(2{+}1)$$ -dimensional...     

Multiplicative dynamical systems in terms of the induced dynamics

Theoretical and Mathematical Physics - We realize an example of induced dynamics using new multiplicative determinant relations whose roots give the particle positions. We present both a general...