Search results for "Integrable systems"
showing 10 items of 256 documents
Integrable systems and moduli spaces of curves
2016
This document has the purpose of presenting in an organic way my research on integrable systems originating from the geometry of moduli spaces of curves, with applications to Gromov-Witten theory and mirror symmetry. The text contains a short introduction to the main ideas and prerequisites of the subject from geometry and mathematical physics, followed by a synthetic review of some of my papers (listed below) starting from my PhD thesis (October 2008), and with some open questions and future developements. My results include: • the triple mirror symmetry among P 1-orbifolds with positive Euler characteristic , the Landau-Ginzburg model with superpotential −xyz + x p + y q + z r with 1 p + …
Dissipative solitons for mode-locked lasers
2012
International audience; Dissipative solitons are localized formations of an electromagnetic field that are balanced through an energy exchange with the environment in presence of nonlinearity, dispersion and/or diffraction. Their growing use in the area of passively mode-locked lasers is remarkable: the concept of a dissipative soliton provides an excellent framework for understanding complex pulse dynamics and stimulates innovative cavity designs. Reciprocally, the field of mode-locked lasers serves as an ideal playground for testing the concept of dissipative solitons and revealing their unusual dynamics. This Review provides basic definitions of dissipative solitons, summarizes their imp…
Kondo Resonance in a Mesoscopic Ring Coupled to a Quantum Dot: Exact Results for the Aharonov-Bohm/Casher Effects
2000
We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in a one-dimensional mesoscopic ring coupled to a side-branch quantum dot at Kondo resonance. For privileged values of the Aharonov-Bohm-Casher fluxes, the problem can be mapped onto an integrable model, exactly solvable by a Bethe ansatz. In the case of a pure magnetic Aharonov-Bohm flux, we find that the presence of the quantum dot has no effect on the persistent current. In contrast, the Kondo resonance interferes with the spin-dependent Aharonov-Casher effect to induce a current which, in the strong-coupling limit, is independent of the number of electrons in the ring.
On critical behaviour in systems of Hamiltonian partial differential equations
2013
Abstract We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P $$_I$$ I ) equation or its fourth-order analogue P $$_I^2$$ I 2 . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
Dimension bounds in monotonicity methods for the Helmholtz equation
2019
The article [B. Harrach, V. Pohjola, and M. Salo, Anal. PDE] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering coefficients satisfy $q_1 \leq q_2$, then the corresponding Neumann-to-Dirichlet operators satisfy $\Lambda(q_1) \leq \Lambda(q_2)$ up to a finite-dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if $q_1$ and $q_2$ have the same number of positive Neumann eigenvalues, then the finite-dimensional space is trivial. peerReviewed
Study of single top production at high energy electron positron colliders
2014
The effect of single top production on the study of top quark pair production in future high energy electron--positron colliders is evaluated. The rate of the single top quark production process is sizeable throughout a large range of center-of-mass energies and cannot easily be distinguished from the dominant pair production process. We discuss the impact on the top quark mass extraction from a scan through the pair production threshold and the determination of top quark form factors in the continuum. These results advocate for the exploration of the inclusive $e^+e^-\rightarrow W^+bW^-\bar{b}$ process, that includes both top quark pair and single top quark production.
The Complete Solution of the Classical SL(2,ℝ/U(1) Gauged WZNW Field Theory
1998
We prove that any gauged WZNW model has a Lax pair representation, and give explicitly the general solution of the classical equations of motion of the SL(2,R)/U(1) theory. We calculate the symplectic structure of this solution by solving a differential equation of the Gelfand-Dikii type with initial state conditions at infinity, and transform the canonical physical fields non-locally onto canonical free fields. The results will, finally, be collected in a local B\"acklund transformation. These calculations prepare the theory for an exact canonical quantization.
Abelian current algebra and the Virasoro algebra on the lattice
1993
We describe how a natural lattice analogue of the abelian current algebra combined with free discrete time dynamics gives rise to the lattice Virasoro algebra and corresponding hierarchy of conservation laws.
Supersymmetry in non commutative superspaces
2003
Non commutative superspaces can be introduced as the Moyal-Weyl quantization of a Poisson bracket for classical superfields. Different deformations are studied corresponding to constant background fields in string theory. Supersymmetric and non supersymmetric deformations can be defined, depending on the differential operators used to define the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N>1), some new deformations can be defined, with no analogue in the N=1 case.
Lorentz harmonics and superfield action. D=10, N=1 superstring
2000
We propose a new version of the superfield action for a closed D=10, N=1 superstring where the Lorentz harmonics are used as auxiliary superfields. The incorporation of Lorentz harmonics into the superfield action makes possible to obtain superfield constraints of the induced worldsheet supergravity as equations of motion. Moreover, it becomes evident that a so-called 'Wess-Zumino part' of the superfield action is basically a Lagrangian form of the generalized action principle. We propose to use the second Noether theorem to handle the essential terms in the transformation lows of hidden gauge symmetries, which remove dynamical degrees of freedom from the Lagrange multiplier superfield.