On the Treves theorem for the AKNS equation

According to a theorem of Treves, the conserved functionals of the AKNS equation vanish on all pairs of formal Laurent series of a specified form, both of them with a pole of the first order. We propose a new and very simple proof for this statement, based on the theory of B\"acklund transformations; using the same method, we prove that the AKNS conserved functionals vanish on other pairs of Laurent series. The spirit is the same of our previous paper on the Treves theorem for the KdV, with some non trivial technical differences.Comment: LaTeX, 16 page

Boundary RG Flow Associated with the AKNS Soliton Hierarchy

We introduce and study an integrable boundary flow possessing an infinite number of conserving charges which can be thought of as quantum counterparts of the Ablowitz, Kaup, Newell and Segur Hamiltonians. We propose an exact expression for overlap amplitudes of the boundary state with all primary states in terms of solutions of certain ordinary linear differential equation. The boundary flow is terminated at a nontrivial infrared fixed point. We identify a form of whole boundary state corresponding to this fixed point.Comment: 54 page

On the equivalence of two super Korteweg\u2013deVries theories: A bi\u2010Hamiltonian viewpoint

From a bi-Hamiltonian viewpoint the equivalence of two supersymmetric Korteweg-deVries theories, introduced by Manin-Radul and Laberge-Mathieu, is discussed herein. It is shown that the transformation connecting the two theories (proposed recently in the literature) preserves the bi-Hamiltonian structures; moreover, another derivation of this transformation, stemming from bi-Hamiltonian reduction theory and strongly emphasizing the geometrical meaning of the above equivalence, is presented

On the constants in some inequalities for the Sobolev norms and pointwise product

We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities

On the constants in a basic inequality for the Euler and Navier-Stokes equations

We consider the incompressible Euler or Navier-Stokes (NS) equations on a d- dimensional torus T^d; the quadratic term in these equations arises from the bilinear map sending two velocity fields v,w : T^d \u2192 R^d into v . Dw, and also involves the Leray projection L onto the space of divergence free vector fields. We derive upper and lower bounds for the constants in some inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants K_{n d} 61 K_n in the basic inequality ||L(v . Dw)||_n <= K_n || v ||_n || w ||_{n+1}, where n 08 (d/2,+ 1e) and v,w are in the Sobolev spaces H^n, H^{n+1} of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d = 3 and some values of n. Some practical motivations are indicated for an accurate analysis of the constants K_n

On the average principle for one-frequency systems

We consider a perturbed integrable system with one frequency, and the approximate dynamics for the actions given by averaging over the angle. The classical theory grants that, for a perturbation of order epsilon, the error of this approximation is O(epsilon) on a time scale O(1/epsilon), for epsilon -> 0. We replace this qualitative statement with a fully quantitative estimate; in certain cases, our approach also gives a reliable error estimate on time scales larger than 1/epsilon. A number of examples are presented; in many cases our estimator practically coincides with the envelope of the rapidly oscillating distance between the actions of the perturbed and of the averaged systems. Fairly good results are also obtained in some "resonant" cases, where the angular frequency is small along the trajectory of the system. Even though our estimates are proved theoretically, their computation in specific applications typically requires the numerical solution of a system of differential equations. However, the time scale for this system is smaller by a factor epsilon than the time scale for the perturbed system. For this reason, computation of our estimator is faster than the direct numerical solution of the perturbed system; the estimator is rapidly found also in cases when the time scale makes impossible (within reasonable CPU times) or unreliable the direct solution of the perturbed system

Bihamiltonian reduction and susy KdV

A terse account is given of the bihamiltonian reduction scheme for supersymmetric evolution equations, which are viewed as defining Hamiltonian vector fields on supermanifolds. As a working example, the Manin-Radul susy KdV is considered in detail

On the continuous limit of integrable lattices II. Volterra systems and sp(N) theories

A connection is suggested between the zero-spacing limit of a generalized N-fields Volterra (V_N) lattice and the KdV-type theory which is associated, in the Drinfeld-Sokolov classification, to the simple Lie algebra sp(N). As a preliminary step, the results of the previous paper [Morosi and Pizzocchero, Commun. Math. Phys. 1996] are suitably reformulated and identified as the realization for N = 1 of the general scheme proposed here. Subsequently, the case N = 2 is analyzed in full detail; the infinitely many commuting vector fields of the V_2 system (with their Hamiltonian structure and Lax formulation) are shown to give in the continuous limit the homologous sp(2) KdV objects, through conveniently specified operations of field rescaling and recombination. Finally, the case of arbitrary N is attacked, showing how to obtain the sp(N) Lax operator from the continuous limit of the V_N system