Volume 191, Nº 3 (2017)

We continue the previously started study of the development of a direct method for constructing the Lax pair for a given integrable equation. This approach does not require any addition assumptions about the properties of the equation. As one equation of the Lax pair, we take the linearization of the considered nonlinear equation, and the second equation of the pair is related to its generalized invariant manifold. The problem of seeking the second equation reduces to simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is remarkable that the second equation of this pair allows easily finding a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs thus obtained differ from usual ones in having a higher order or a higher matrix dimensionality. We show with examples that they reduce to the usual pairs by reducing their order. As an example, we consider an integrable double discrete system of exponential type and its higher symmetry for which we give the Lax pair and construct the conservation laws.

We show that in a four-dimensional space–time a complex scalar field can be associated with a one-dimensionally extended object, called a charged string. The string is said to be charged because the complex scalar field describing it interacts with an electromagnetic field. A charged string is characterized by an extension of the symmetry group of the charge space to a group of stretch rotations. We propose relativistically invariant and gauge-invariant equations describing the interaction of a complex scalar field with an electromagnetic field, and each solution of them corresponds to a charged string. We achieve this by introducing the notion of a charged string index, which, as verified, takes only integer values. We establish equations from which it follows that charged strings fit naturally into the framework of the Maxwell–Dirac electrodynamics.

We consider the melting of a two-dimensional system of collapsing hard disks (a system with a hard-disk potential to which a repulsive step is added) for different values of the repulsive-step width. We calculate the system phase diagram by the method of the density functional in crystallization theory using equations of the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young theory to determine the lines of stability with respect to the dissociation of dislocation pairs, which corresponds to the continuous transition from the solid to the hexatic phase. We show that the crystal phase can melt via a continuous transition at low densities (the transition to the hexatic phase) with a subsequent transition from the hexatic phase to the isotropic liquid and via a first-order transition. Using the solution of renormalization group equations with the presence of singular defects (dislocations) in the system taken into account, we consider the influence of the renormalization of the elastic moduli on the form of the phase diagram.

We prove a theorem on the exact asymptotic relations of large deviations of the Bogoliubov measure in the L^pnorm for p = 4, 6, 8, 10 with p > p₀, where p₀ = 2+4π²/β²ω²is a threshold value, β > 0 is the inverse temperature, and ω > 0 is the natural frequency of the harmonic oscillator. For the study, we use the Laplace method in function spaces for Gaussian measures.

We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.