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Quasi-classical approximation in the intrinsic description of the vortex filament dynamics is discussed. Within this approximation, the governing equations are given by elliptic system of quasi-linear partial differential equations of the first order. Dispersionless Da Rios system and dispersionless Hirota equation are among them. They describe motion of vortex filament with slow-varying curvature and torsion without or with axial flow. Gradient catastrophe for governing equations is studied. It is shown that geometrically this catastrophe manifests as a fast oscillation of a filament curve around the rectifying plane which resembles the flutter of airfoils. Analytically, it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painleve I equation.

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is amenable to the quasiclassical DBAR-dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed., Comment: 33 pages, 7 figures

A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles that constitute the simplest pair of reciprocal figures representing both a framework and a self-stress encapsulate the integrable discrete BKP equation and its Schwarzian version. The inherent Mobius invariant nature of the Schwarzian BKP equation is then exploited to define reciprocity in an inversive geometric setting. Integrable pairs of lattices of nontrivial combinatorics consisting of reciprocal triangles and their natural generalizations are discussed. Particular reductions of these BKP lattices are related to the integrable discrete versions of Darboux's (2+1)-dimensional sine-Gordon equation and the classical Tzitzeica equation of affine geometry. Furthermore, it is shown that octahedral figures and their hexahedral reciprocals as considered by Maxwell likewise give rise to discrete integrable systems and associated integrable lattices.

The 2+1 dimensional integrable generalization of the sine-Gordon equation symmetric in the spatial variables is studied by the inverse spectral transform method. The solutions with functional parameters, plane solitons (kinks) and plane breathers are constructed by the dressing method based on the mixed nonlocal -∂-problem. The initial value problem for this equation with the constant boundaries is solved in both cases σ2 = ± 1.

The 2 + 1-modified Kadomtsev-Petviashvili (mKP) equation is studied by the inverse-spectral-transform method. The initial-value problems for the mKP-1 and mKP-11 equations are solved by the nonlocal Riemann-Hilbert and techniques for initial data decaying sufficiently rapidly at infinity. The lump solutions for the mKP-I equation are found explicitly. Wide classes of the exact solutions for the mKP equation—namely, the rational solutions, including the plane lumps for the mKP-I equation; solutions with functional parameters; the plane solitons; and breathers—are constructed by the use of the method based on the nonlocal . The Miura transformation between the mKP and KP equations is discussed.

The Gelfand-Dikij spectral problem is considered in the AKNS-technique framework. The general form of nonlinear evolution equations connected with matrix Gelfand-Dikij spectral problem is found. The infinitedimensional abelian group of general Backlund transformations is constructed. The infinite family of Hamiltonian structures connected with the nonlinear equations under consideration is found.

The transformation properties of the partial differential equations integrable by the inverse scattering transform method are considered. It is shown that the nonlinear transformations characteristic to the integrable equations (symmetry groups, Backlund-transformations) and integrable equations themselves are contained in certain universal nonlinear transformations group which is defined by linear spectral problem.