1,721,012 research outputs found
Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular β -Ensemble and Double Confluent Heun Equation
Full text linkWe consider the discrete defocusing nonlinear Schrodinger equation in its integrable version, which is called defocusing Ablowitz-Ladik lattice. We consider periodic boundary conditions with period N and initial data sampled according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz-Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular beta-ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation
Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach
No full textWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation u(t) + 6uu(x) + epsilon(2)u(xxx) = 0, u(x, t = 0, epsilon) = u(0)(x), for epsilon small, near the point of gradient catastrophe (x(c), t(c)) for the solution of the dispersionless equation u(t) + 6uu(x) = 0. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit
Soliton Shielding of the Focusing Nonlinear Schrödinger Equation
No full textWe first consider a deterministic gas of N solitons for the focusing nonlinear Schrodinger (FNLS) equation in the limit N -infinity with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with the point spectrum the center of the disk. We call this effect soliton shielding. We show that this behavior is robust and survives also for a stochastic soliton gas: indeed, when the N-soliton spectrum is chosen as random variables either uniformly distributed on the circle, or chosen according to the statistics of the eigenvalues of the Ginibre random matrix the phenomenon of soliton shielding persists in the limit N -infinity. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically steplike oscillatory, namely, the initial profile is a periodic elliptic function in the negative x direction while it vanishes exponentially fast in the opposite direction
Exactly Solvable Anharmonic Oscillator, Degenerate Orthogonal Polynomials and Painlevé II
Full text linkUsing WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values of t in the complex plane
for which the spectrum of the quartic anharmonic oscillator in the complex plane with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials, i.e. the poles of rational solutions of the second Painleve equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials
Discrete Integrable Systems and Random Lax Matrices
Full text linkWe study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number N of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian beta-ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz-Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states
Entanglement of Two Disjoint Intervals in Conformal Field Theory and the 2D Coulomb Gas on a Lattice
Full text linkIn the conformal field theories given by the Ising and Dirac models, when the system is in the ground state, the moments of the reduced density matrix of two disjoint intervals and of its partial transpose have been written as partition functions on higher genus Riemann surfaces with symmetry. We show that these partition functions can be expressed as the grand canonical partition functions of the two-dimensional two component classical Coulomb gas on certain circular lattices at specific values of the coupling constant
Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit
Full text linkWe consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed byN ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature β−1. Given a fixed 1 ≤ m ≪ N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order β, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics
On the long time asymptotic behaviour of the modified Korteweg de Vries equation with step-like initial data
Full text linkWe study the long time asymptotic behaviour of the solution q(x,t) of the modified Korteweg de Vries equation (MKdV) with step-like initial datum asymptotic to c_+ at +infinity and to c_- at -infinity. We show that the solution for long times decomposes in the (x,t) plane in three main regions:1. a region where solitons and breathers travel with positive velocities on a constant background c_+;2. an expanding oscillatory region {\color{black} (that generically contains breathers)};3. a region of breathers travelling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the step initial data. The phase shift depends on the solitons, breathers and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers and radiation
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