This paper is a brief review of the program initiated some years ago in [JMP]. The main goal of this program is to understand the formation and the propagation properties of surface states in regions with corrugated boundaries. The history of this problem and its physical aspects are discussed in [JMP, KP]. The results described in this review are obtained in collaborations with Yoram Last and ...

In 70’s there was discovered a construction how to attach to some algebraicgeometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. The construction was successfully used in the theory of integrable systems, particularly, for the KP and KdV equations [10, 19]. There were also found some applications to the moduli of algebraic curves [2, 3]. Now it is know...

We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m)-minimal superconformal models coupled to 2D-supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all...

Previous studies have explored the use of departure from Hardy-Weinberg equilibrium (DHW) for fine mapping Mendelian disorders and for general fine mapping. Other studies have used Hardy-Weinberg tests for genotyping quality control. To enable investigators to make rational decisions about whether DHW is due to genotyping error or to underlying biology, we developed an analytic framework and so...

The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of RiemannHilbert problem. This is however still short of clarifying geometric ingredients of twistor theory, such as twistor lines and twistor surfaces. A more geometric approach can be developed in a Hamilton-Jacobi formalism of Gibbons and Kodama. A...

The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov–Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Wh...

The main results for the two-dimensional quantum gravity, conjectured from the matrix model or integrable approach, are presented in the form to be compared with the world-sheet or Liouville approach. In spherical limit the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials, based on residue formulas from quasiclassical hierarchi...

Using gauge transformations for the corresponding generating pseudo-differential operators L in terms of eigenfunctions and adjoint eigenfunctions, we construct several types of auto-Bäcklund transformations for the KP hierarchy with self-consistent sources (KPHSCS) and mKP hierarchy with self-consistent sources (mKPHSCS) respectively. The Bäcklund transformations from the KPHSCS to mKPHSCS are...

We present a new class of topological conformal eld theories (TCFT) characterized by a rational W potential, which includes the minimal models of A and D types as its subclasses. An explicit form of the W potential is found by solving the underlying dispersionless KP hierarchy in a particular small phase space. We discuss also the dispersionless KP hierarchy in large phase spaces by reformulati...

We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear 2× 2 system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the Schlesinger system and the sixth Painlevé equation is given from this linear system. A unified approach to treat the other Painlevé equations from the usual two-component KP...