Том 193, № 3 (2017)

We develop a previously proposed algebraic technique for a Hamiltonian approach to evolution systems of partial differential equations including constrained systems and propose a defining system of equations (suitable for computer calculations) characterizing the Hamiltonian operators of a given form. We demonstrate the technique with a simple example.

We consider the simplest Fuchsian second-order equations with particular attention to the role of apparent singularities. We show the relation to the Painlevé equation and follow the matrix formulation of the problem.

Using the bilinear transformation method, we derive general rogue-wave solutions of the Zakharov equation. We present these Nth-order rogue-wave solutions explicitly in terms of Nth-order determinants whose matrix elements have simple expressions. We show that the fundamental rogue wave is a line rogue wave with a line profile on the plane (x, y) arising from a constant background at t ≪ 0 and then gradually tending to the constant background for t ≫ 0. Higher-order rogue waves arising from a constant background and later disappearing into it describe the interaction of several fundamental line rogue waves. We also consider different structures of higher-order rogue waves. We present differences between rogue waves of the Zakharov equation and of the first type of the Davey–Stewartson equation analytically and graphically.

We consider the critical nonunitary minimal model M(3, 5) with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz (TBA) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated A₄restricted solid-on-solid Forrester–Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the (r, s) = (1, 1) sector and then determine their corresponding energies. We classify the excitations in terms of (m, n) systems.

We analyze how the compositeness of a system affects the characteristic time of equilibration. We study the dynamics of open composite quantum systems strongly coupled to the environment after a quantum perturbation accompanied by nonequilibrium heating. We use a holographic description of the evolution of entanglement entropy. The nonsmooth character of the evolution with holographic entanglement is a general feature of composite systems, which demonstrate a dynamical change of topology in the bulk space and a jumplike velocity change of entanglement entropy propagation. Moreover, the number of jumps depends on the system configuration and especially on the number of composite parts. The evolution of the mutual information of two composite systems inherits these jumps. We present a detailed study of the mutual information for two subsystems with one of them being bipartite. We find five qualitatively different types of behavior of the mutual information dynamics and indicate the corresponding regions of the system parameters.

Using the method of diagram techniques for the spin and Fermi operators in the framework of the SU(2)-invariant spin-fermion model of the electron structure of the CuO₂plane of copper oxides, we obtain an exact representation of the Matsubara Green’s function D_⊥(k, iω_m) of the subsystem of localized spins. This representation includes the Larkin mass operator Σ_L(k, iω_m) and the strength and polarization operators P(k, iω_m) and Π(k, iω_m). The calculation in the one-loop approximation of the mass and strength operators for the Heisenberg spin system in the quantum spin-liquid state allows writing the Green’s function D_⊥(k, iω_m) explicitly and establishing a relation to the result of Shimahara and Takada. An essential point in the developed approach is taking the spin-polaron nature of the Fermi quasiparticles in the spin-fermion model into account in finding the contribution of oxygen holes to the spin response in terms of the polarization operator Π(k, iω_m).

We consider a p-adic solid-on-solid (SOS) model with a nearest-neighbor coupling, m+1 spins, and a coupling constant J ∈ Q_pon a Cayley tree. We find conditions under which a phase transition does not occur in the model. We show that if p | m + 1 for some J, then a phase transition occurs. Moreover, we formulate a criterion for the boundedness of p-adic Gibbs measures for the (m+1)-state SOS model.