On the soliton solutions of a family of Tzitzeica equations

We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. The first type corresponds to a set of 6 symmetrically situated discrete eigenvalues of the Lax operator $L$; to each soliton of the second type one relates a set of 12 discrete eigenvalues of $L$. We also outline how one can construct general $N$ soliton solution containing $N_1$ solitons of first type and $N_2$ solitons of second type, $N=N_1+N_2$. The possible singularities of the solitons and the effects of change of variables that relate the different members of Tzitzeica family equations are briefly discussed. All equations allow quasi-regular as well as singular soliton solutions.Comment: 24 pages, 1 figur

On a discrete analog of the Tzitzeica equation

A discrete analog of the Tzitzeica equation is found in the form of quad-equation. Its continuous symmetry is an inhomogeneous Narita--Bogoyavlensky type lattice equation which defines a discretization of the Sawada--Kotera equation. The integrability of these discretizations is proven by construction of the Lax representations.Comment: 13 p

Abelian vortices from Sinh--Gordon and Tzitzeica equations

It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1) vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh--Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlev\'e transcendents are used to compute explicit values for the strength of the vortices.Comment: 10 pages. Possible physical applications discussed + one reference added. Final version, to appear in Physics Letters

Killing fields and Conservation Laws for rank-1 Toda field equations

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzeica equation, completing the work in Fox and Goertsches (2011). We define a notion of finite-type for integral manifolds of exterior differential systems directly in terms of conservation laws that generalizes the definition of Pinkall-Sterling (1989). The definition applies to any exterior differential system that has infinitely many conservation laws possessing a normal form. Finally, we show that, for the rank-one Toda field equations, every characteristic cohomology class has a translation invariant representative as an undifferentiated conservation law. Therefore the characteristic cohomology defines de Rham cohomology classes on doubly periodic solutions.Comment: 26 page

On a new avatar of the sine-Gordon equation

A chain of transformations is found which relates one new integrable case of the generalized short pulse equation of Hone, Novikov and Wang [arXiv:1612.02481] with the sine-Gordon equation.Comment: 10 page

Minimal tori in five-dimensional sphere in C3C^3

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.Comment: AmSTeX, 10 pages, amsppt styl

Surfaces in Lie sphere geometry and the stationary Davey-Stewartson hierarchy

We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces we derive the stationary modified Veselov-Novikov equation, whose role in the theory of these surfaces is similar to that of Calapso's equation in the theory of isothermic surfaces. Since Calapso's equation itself turns out to be related to the stationary Davey-Stewartson equation, these results shed some new light on differential geometry of the stationary Davey-Stewartson hierarchy. Diagonally-cyclidic surfaces are the natural Lie sphere analogs of the isothermally-asymptotic surfaces in projective differential geometry for which we also derive the stationary modified Veselov-Novikov equation with the different real reduction. Parallels between invariants of surfaces in Lie sphere geometry and reciprocal invariants of hydrodynamic type systems are drawn in the conclusion.Comment: Latex, 31 page

Symmetry groups and Lagrangians associated to Tzitzeica surfaces

One applies the symmetry group theory for study the partial differential equations of Tzitzeica surfaces theory. One finds infinitesimal symmetries, Lagrangians and a new solution of Titzeica equation.Comment: 21 page

Formal Killing fields for minimal Lagrangian surfaces in complex space forms

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the differential algebraic inductive formulas for a pair of loop algebra $\mathfrak{sl}(3,\mathbb{C})[[\lambda]]$-valued canonical formal Killing fields. As a result, we give a complete classification of the (infinite sequence of) Jacobi fields for the minimal Lagrangian system. We also obtain an infinite sequence of higher-order conservation laws from the components of the formal Killing fields.Comment: 57 pages. v4. misprints are correcte

Solitons on a finite-gap background in Bullough-Dodd-Jiber-Shabat model

The determinant formula for N-soliton solutions of the Bullough-Dodd-Jiber-Shabat equation on a finite-gap background is obtained. Nonsingularity conditions for them and their asymptotics are investigated.Comment: AmSTeX, 7 pages, amsppt style, 1 figur