Nonlinear Sciences

We propose a new type of soliton equation, which is obtained from the generalized discrete BKP equation. The obtained equation admits two types of soliton solutions. The signs of amplitude and velocity of the soliton solution are opposite to the other. We also propose the ultradiscrete analogues of them. The ultradiscrete equation also admits the similar properties. In particular it behaves the original Box-Ball system in a special case.

The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been suggested earlier in \cite{GHY} which coincide with the corresponding Drinfeld-Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been approved by numerous examples. In the present article we study those systems from this class which are related to the algebras A (1) N−1 . We found the Lax pair for arbitrary N , briefly discussed the possibility of using the method of formal diagonalization of Lax operators for describing a series of local conservation laws and illustrated the technique using the example of N=3 . Higher symmetries of the system A (1) N−1 are presented in both characteristic directions. Found recursion operator for N=3 . It is interesting to note that this operator is not weakly nonlocal.

We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.

We investigate the nonisospectral effects of a semi-discrete nonlinear Schrödinger equation, which is a direct integrable discretisation of its continuous counterpart. Bilinear form and double casoratian solution of the equation are presented. Dynamics of solutions are analyzed. Both solitons and multiple pole solutions admit space-time localized rogue wave behavior. And more interestingly, the solutions allow blow-up at finite time t .

Abelian integrals arise in the mathematical description of various physical processes. According to Abel's theorem these integrals are related to motion of a set of points along a plane curve around fixed points, which are relatively little used in physics applications. We propose to interpret coordinates of the fixed points either as parameters of exact discretization or as additional first integrals for equations of motion reduced to Abelian quadratures.

For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.

Lie symmetry algebra of the dispersionless Davey-Stewartson (dDS) system is shown to be infinite-dimensional. The structure of the algebra turns out to be Kac-Moody-Virasoro one, which is typical for integrable evolution equations in 2+1 -dimensions. Symmetry group transformations are constructed using a direct (global) approach. They are split into both connected and discrete ones. Several exact solutions are obtained as an application of the symmetry properties.

This paper is a continuation of our previous work in which we studied a dispersionless limits of some integrable spin systems (magnetic equations). Now, we shall present dispersionless limits of some integrable generalized Heisenberg ferromagnet equations.

This is a write-up of the lectures on dispersionless equations in 1+1, 2+1 and 3+1 dimensions presented by one of us (RM) at "Eurasian International Center for Theoretical Physics" (EICTP). We provide pedagogical introduction to the subject and summarize well-known results and some recent developments in theory of integrable dispersionless equations. Almost all results presented are available in the literatures. We just add some new results related to dispersionless limits of integrable magnetic equations. On the contrary, some of the basic tools of the integrable dispersionless systems and related theoretical techniques are not available in a pedagogical format. For this reason we think that it seems worthwhile to present basics of theory of dispersionless equations here for the benefit of beginner students. We begin with a detailed exposition of the well-known dispersionless systems like dKdVE, dNLSE, dKPE, dDSE and other classical soliton equations. We present in detail some new dispersionless systems. Next we develop the dispersionless limits of known magnetic equations. Lastly, we discuss in full detail the Lax representation formulation of some presented dispersionless equations. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field of integrable dispersionless equations and related topics.

We derive a dispersionless integrable system describing a local form of a general three-dimensional Einstein-Weyl geometry with an Euclidean (positive) signature, construct its matrix extension and demonstrate that it leads to the Bogomolny equations for a non-abelian monopole on an Einstein-Weyl geometry background. The corresponding dispersionless integrable hierarchy, its matrix extension and the dressing scheme are also considered.