Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces e cient algorithms exist even for almost degenerate surfaces. This allows the numeri...

and Applied Analysis 3 P(X, Y) = ∑ m i=0 a i (X)Y i is an irreducible polynomial in the complex domain C[X, Y] such that P [X (ξ) , Y (ξ)] = m ∑ i=0 a i (X (ξ)) Y i (ξ) = 0, (13) where a i (X), (i = 0, 1, 2, . . . , m) are polynomials of X and a m (X) ̸ = 0. Equation (13) is called the first integral to (12a) and (12b). Due to the Division Theorem, there exists a polynomial h(X) + g(X)Y in the c...

This work consist of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups G with central extensions, with gauge group H being certain (infinite-dimensional) subgroup of G. We show that there exist generalized “zero-curvature” representation of the pertinent equations of motion on the ...

Recent studies on (2 + 1 )-dimensional integrable systems have shown the existence of solutions describing localized coherent structures with remarkable dynamical properties.‘*2 Unlike one-dimensional solitons these structures do not preserve, in general, their form upon interaction2 and may undergo processes of fusion and fission.3p4 The basic model exhibiting these solutions is the DaveyStewa...

In this paper, we show that various noncommutative integrable equations can be derived from noncommutative anti-self-dual Yang-Mills equations in the split signature, which include noncommutative versions of Korteweg-de Vries, Non-Linear Schrödinger, N -wave, Davey-Stewartson and Kadomtsev-Petviashvili equations. U(1) part of gauge groups for the original Yang-Mills equations play crucial roles...

The Weierstrass representation for surfaces in R3 [8, 13] was generalized for surfaces in R4 in [12] (see also [4]). This paper uses the quaternion language and the explicit formulas for such a representation were written by Konopelchenko in [9] for constructing surfaces which admit soliton deformation governed by the Davey–Stewartson equations. This generalizes his results from [8] where he in...