1,721,011 research outputs found
Semilinear evolution equations in Frechet spaces, Quaderno 23/1999, Dipartimento di Matematica dell'Universita' di Milano
No full textThis work is devoted to the Cauchy problem for a class of semilinear
evolution equations in abstract Frechet spaces. Existence of the solution is
derived with the Peano-Picard method; uniqueness and smoothness of the
flow are also proved. Criteria are given for the existence of global solutions,
defined for all positive times. Also, approximate solutions and the related
errors are discussed. The proposed framework can be applied to the nonlinear
Schroedinger, Klein-Gordon and heat equations, formulated in appropriate
spaces of smooth functions; the general theory controls the derivatives of any
order of the solutions
On the continuous limit of integrable lattices III. Kupershmidt systems and sl(n+1) KdV theories
No full textWe discuss the connection between the zero-spacing limit of the N- fields Kupershmidt lattice and the KdV-type theory corresponding to the Lie algebra sl(N+1). The case N = 2 is worked out in detail, recovering from the limit process the Boussinesq theory with its infinitely many commuting vector fields, their Lax pairs and Hamiltonian formulations. Actually, the ‘recombination method’ proposed here to derive the Boussinesq hierarchy from the limit of the N = 2 Kupershmidt system works, in principle, for arbitrary N
On the continuous limit of integrable lattices I. The KM lattice and KdV theory
No full textFrom the MR review by Malcolm Adams
" It has been known for some time that the Korteweg-de Vries (KdV) equation can be obtained as a continuous limit of the Kac-Moerbeke (KM) lattice system, an integrable system on a one-dimensional lattice of particles related to the Toda lattice [M. Schwarz, Jr., Adv. in Math. 44 (1982), no. 2, 132--154; MR0658538 (83i:35152)]. Although that work helps to give insight into the structure of the KdV equation, it does not show that the entire KdV hierarchy of commuting evolutionary partial differential equations can be obtained in the same limiting fashion."
The present paper fills that gap by giving a systematic construction of the entire KdV theory as a continuous limit of the KM theory. The approach is first to give the correspondence between the bi-Hamiltonian structures of the two theories and then to produce the corresponding hierarchies through recursion operators. The paper is well written and fairly self-contained, giving a nice brief overview of KM theory
On the equivalence of two sKdV theories: a biHamiltonian viewpoint
No full textFrom the MR review by M.Mehdi: "The authors are interested in the equivalence of the sKdV theory introduced by Manin and Radul (MR) and the so-called N=2, α = −2 sKdV theory of Laberge and Mathieu (LM). They use the approach of Inami and Kanno to obtain from a Lie superalgebraic framework the matrix and scalar Lax formalism for sKdV theories, and subsequently to infer a Hamiltonian formalism by reducing the R-matrix Poisson quadratic structure of the algebra of superpseudodifferential operators. The authors obtain directly a bi-Hamiltonian structure reducing two general Poisson structures associated to the loop superalgebras of simple Lie superalgebras. They find a transformation between the phase spaces of the MR and LM theories, setting up an equivalence between the Lax formulation and mapping the former hierarchy into the latter. They prove that the Inami-Kanno transformation preserves the bi-Hamiltonian structures corresponding to the MR and LM theories.
On the Euler equation: bi-Hamiltonian structures and integrals in involution
No full textWe propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the `physical' space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion
On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach
No full textWe give a Lie superalgebraic interpretation of the biHamiltonian structure of the known susy KdV equations. We show that he loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the susy KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra
A fully supersymmetric AKNS theory
No full textWe construct a fully supesymmetric biHamiltonian theory in four superfields, admitting zero curvature and Lax formulation. This theory is an extension of the classical AKNS, which can be recovered as a reduction. Other supersymmetric theories are obtained as reductions of the susy AKNS, namely a nonlinear Scroedinger, a modified KdV and the Manin-Radul KdV. The susy nonlinear Schroedinger hierarchy is related to the one of Roelofs and Kersten; we determine its bihamiltonian and Lax formulation. Finally, we show that the susy KdV's mentioned before are related through a susy Miura map
On the constants in a basic inequality for the Euler and Navier-Stokes equations
No full textWe consider the incompressible Euler or Navier-Stokes equations on a d-dimensional torus ; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w into v . w’, 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 Kn in the basic inequality |L(v . w’) |_n <= Kn | v|_n | w |_{n+1}, where
n > d/2 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 Kn, making reference to other works on the approximate solutions of Euler or Navier-Stokes equations
On the biHamiltonian interpretation of the Lax formalism
No full textWe propose a general framework for constructing systematically the Lax formulation of the soliton equation using the bi-Hamiltonian formalism. The method is applied to several examples, both classical and supersymmetric
On the continuous limit of integrable lattices II. Volterra systems and sp(N) theories
No full textA 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). The recombination method developed in a previous paper of ours is applied to study in full detail the case N=2: the infinitely many commuting vector fields, the Hamiltonian structure and the Lax formulation of the corresponding Volterra system V_2 are shown to give in the continuous limit the homologous sp(2) KdV objects, through 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
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