On τ -function of conjugate nets
نویسنده
چکیده
We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τ -function. We investigate the potential using the non-local ∂̄ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally, we give some arguments extending that interpretation to multicomponent Kadomtsev–Petviashvili hierarchy. Dedicated to F. Calogero for his 70-th birthday.
منابع مشابه
Charged Free Fermions, Vertex Operators and Classical Theory of Conjugate Nets
We show that the quantum field theoretical formulation of the τ function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic vertex operators are Lévy and adjoint Lévy transformations and iii) the diagonal s...
متن کاملExistence of Subharmonic Periodic Solutions to a Class of Second-Order Non-Autonomous Neutral Functional Differential Equations
and Applied Analysis 3 Let us consider the functional I x defined on H1 0 0, 2γτ by I x ∫2γτ 0 [ x′ t x′ t − τ − F t, x t , x t − τ dt. 2.4 For all x, y ∈ H1 0 0, 2γτ and ε > 0, we know that I ( x εy ) I x ε (∫2γτ 0 [ x′ t y′ t − τ x′ t − τ y′ t −Ft, x t εy t , x t − τ εy t − τ ) − F t, x t , x t − τ dt ) ε2 ∫2γτ 0 y′ t y′ t − τ dt. 2.5 It is then easy to see that 〈 I ′ x , y 〉 ∫2γτ 0 [ x′ t y′...
متن کاملConjugate Operators for Finite Maximal Subdiagonal Algebras
Let M be a von Neumann algebra with a faithful normal trace τ , and let H∞ be a finite, maximal, subdiagonal algebra of M. Fundamental theorems on conjugate functions for weak∗-Dirichlet algebras are shown to be valid for noncommutative H∞. In particular the conjugation operator is shown to be a bounded linear map from Lp(M, τ) into Lp(M, τ) for 1 < p < ∞, and to be a continuous map from L1(M, ...
متن کاملModeling of Continuous Systems Using Modified Petri Nets
Due to the changes which may occur in their parameters, systems are usually demonstrated by some subsystems for different conditions. This paper employs Modified Petri Nets (MPN) to model theses subsystems and makes it simple to analyze them. In this method, first, the continuous transfer function is converted to a discrete transfer function and then, by MPN, system is modeled and analyzed. All...
متن کاملτ-Equivalences for analysis of concurrent systems modelled by Petri nets with silent transitions
The paper is devoted to the investigation of behavioural equivalences for Petri nets with silent transitions (τ -equivalences). Basic τ -equivalences and back-forth τ -bisimulation equivalences are supplemented by new ones, giving rise to the complete set of equivalence notions in interleaving / true concurrency and linear / branching time semantics. Their interrelations are examined for the ge...
متن کامل