Search results for "Integrable systems"
showing 10 items of 256 documents
Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
2017
International audience; We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the 'energy' parameter E. We show that as |E| -> infinity, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when |E| is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.
Covariant Operator Formalism for Quantized Superfields
1988
The Takahashi-Umezawa method of deriving the free covariant quantization relations from the linear equations of motion is extended to superfields. The Cauchy problem for free superfields is solved, and an expression for the time independent scalar product is given. For the case of interacting fields, we give the general Kallen-Lehmann spectral representation for the two-point superfield Green functions and, after the introduction of the asymptotic condition for superfields, we give the superfield extension of the Yang-Feldman equation. The case of the D = 2 real scalar superfield and the case of the D = 4 chiral superfield are discussed in detail.
Wronskian and Casorati determinant representations for Darboux–Pöschl–Teller potentials and their difference extensions
2009
We consider some special reductions of generic Darboux?Crum dressing formulae and of their difference versions. As a matter of fact, we obtain some new formulae for Darboux?P?schl?Teller (DPT) potentials by means of Wronskian determinants. For their difference deformations (called DDPT-I and DDPT-II potentials) and the related eigenfunctions, we obtain new formulae described by the ratios of Casorati determinants given by the functional difference generalization of the Darboux?Crum dressing formula.
Other 2N− 2 parameters solutions of the NLS equation and 2N+ 1 highest amplitude of the modulus of theNth order AP breather
2015
In this paper, we construct new deformations of the Akhmediev-Peregrine (AP) breather of order N (or APN breather) with real parameters. Other families of quasirational solutions of the nonlinear Schrodinger (NLS) equation are obtained. We evaluate the highest amplitude of the modulus of the AP breather of order N; we give the proof that the highest amplitude of the APN breather is equal to . We get new formulas for the solutions of the NLS equation, which are different from these already given in previous works. New solutions for the order 8 and their deformations according to the parameters are explicitly given. We simultaneously get triangular configurations and isolated rings. Moreover,…
Breathers and solitons of generalized nonlinear Schrödinger equations as degenerations of algebro-geometric solutions
2011
We present new solutions in terms of elementary functions of the multi-component nonlinear Schr\"odinger equations and known solutions of the Davey-Stewartson equations such as multi-soliton, breather, dromion and lump solutions. These solutions are given in a simple determinantal form and are obtained as limiting cases in suitable degenerations of previously derived algebro-geometric solutions. In particular we present for the first time breather and rational breather solutions of the multi-component nonlinear Schr\"odinger equations.
A form factor approach to the asymptotic behavior of correlation functions in critical models
2011
We propose a form factor approach for the computation of the large distance asymptotic behavior of correlation functions in quantum critical (integrable) models. In the large distance regime we reduce the summation over all excited states to one over the particle/hole excitations lying on the Fermi surface in the thermodynamic limit. We compute these sums, over the so-called critical form factors, exactly. Thus we obtain the leading large distance behavior of each oscillating harmonic of the correlation function asymptotic expansion, including the corresponding amplitudes. Our method is applicable to a wide variety of integrable models and yields precisely the results stemming from the Lutt…
Form factor approach to dynamical correlation functions in critical models
2012
We develop a form factor approach to the study of dynamical correlation functions of quantum integrable models in the critical regime. As an example, we consider the quantum non-linear Schr\"odinger model. We derive long-distance/long-time asymptotic behavior of various two-point functions of this model. We also compute edge exponents and amplitudes characterizing the power-law behavior of dynamical response functions on the particle/hole excitation thresholds. These last results confirm predictions based on the non-linear Luttinger liquid method. Our results rely on a first principles derivation, based on the microscopic analysis of the model, without invoking, at any stage, some correspon…
Thermodynamic limit of particle-hole form factors in the massless XXZ Heisenberg chain
2010
We study the thermodynamic limit of the particle-hole form factors of the XXZ Heisenberg chain in the massless regime. We show that, in this limit, such form factors decrease as an explicitly computed power-law in the system-size. Moreover, the corresponding amplitudes can be obtained as a product of a "smooth" and a "discrete" part: the former depends continuously on the rapidities of the particles and holes, whereas the latter has an additional explicit dependence on the set of integer numbers that label each excited state in the associated logarithmic Bethe equations. We also show that special form factors corresponding to zero-energy excitations lying on the Fermi surface decrease as a …
Microscopic approach to a class of 1D quantum critical models
2015
Starting from the finite volume form factors of local operators, we show how and under which hypothesis the $c=1$ free boson conformal field theory in two-dimensions emerges as an effective theory governing the large-distance regime of multi-point correlation functions in a large class of one dimensional massless quantum Hamiltonians. In our approach, in the large-distance critical regime, the local operators of the initial model are represented by well suited vertex operators associated to the free boson model. This provides an effective field theoretic description of the large distance behaviour of correlation functions in 1D quantum critical models. We develop this description starting f…
Surface free energy of the open XXZ spin-1/2 chain
2012
We study the boundary free energy of the XXZ spin-$\tf{1}{2}$ chain subject to diagonal boundary fields. We first show that the representation for its finite Trotter number approximant obtained by Bortz, Frahm and G\"{o}hmann is related to the partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya determinant representation for the latter quantity we are able to take the infinite Trotter number limit. This yields a representation for the surface free energy which involves the solution of the non-linear integral equation that governs the thermodynamics of the XXZ spin-1/2 chain subject to periodic boundary conditions. We show that this integral representati…