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Register a new userClassification of 3-dimensional integrable scalar discrete equations
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We classify all integrable 3-dimensional scalar discrete quasilinear equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An equation Q=0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the 4-dimensional lattice. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2 has been made.) (Version 3: A few small corrections: one more reference added, the main statement stated more explicitly.)
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We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of equations is understood as 3D-consistency. The latter is a possibility to consistently impose equations of the same type on all the faces of a three-dimensional cube. This allows to set these equations also on multidimensional lattices Z^N. We classify integrable equations with complex fields x, and Q affine-linear with respect to all arguments. The method is based on analysis of singular solutions.
We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. The recent paper cite{FMN} provides a complete list of integrable third order equations of this kind. Here we extend the classification to fifth order equations. Besides the known examples of Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations, as well as fifth order analogues and modifications thereof, our list contains a number of equations which are apparently new. We conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality $w$. The classification procedure consists of two steps. First, we classify quasilinear systems which may (potentially) occur as dispersionless limits of integrable scalar evolutionary equations. After that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic reductions of the dispersionless limit by the full dispersive equation.
We propose an integrable discrete model of one-dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection-diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self-adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naive discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.
Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDAs) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed scheme of deformations.
In the series of recent publications we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are `inherited by the dispersive equations. In this paper we extend this to the fully discrete case. Our only constraint is that the initial ansatz possesses a non-degenerate dispersionless limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.
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