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On approximation of solutions to some \((2 + 1)\)-dimensional integrable systems by Bäcklund transformations. (English. Russian original) Zbl 0952.37050
Sib. Math. J. 41, No. 3, 442-452 (2000); translation from Sib. Mat. Zh. 41, No. 3, 541-553 (2000).
The author studies two examples of \((2 + 1)\)-dimensional Bäcklund transformations of geometric origin: Moutard and Ribaucour transformations. The corresponding Moutard equation \[ u_{xy} = M(x,y)u,\quad u = u(x,y), \] comes from the classical differential geometry, it has many applications to the theory of integrable \((2 + 1)\)-dimensional nonlinear systems. The theory of such systems is used in the theory of integrable systems of hydrodynamical type.
The author gives a positive answer to the question of (local) denseness of solutions obtained from an arbitrary initial solution in the space of all smooth solutions to the equations. It is proven that the classical Moutard and Ribaucour transformations enable us to receive “almost all” solutions to these equations; more exactly, the set of solutions obtained from a given “initial” solution by \(N\) Moutard or Ribaucour transformations for the equations is dense in the space of \(k\)-jets for a sufficiently large \(N\). This allows us to regard the solutions to equations describing triorthogonal curvilinear coordinate systems as completely integrable \((2 + 1)\)-dimensional nonlinear systems. The \(k\)th potential in the chain of Moutard transformations depends (a priori) on the choice of \(2k\) functions of one variable, the initial data for the solutions \(R_s(x,y)\) of the corresponding equations.

MSC:
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35J10 Schrödinger operator, Schrödinger equation
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:
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