192 research outputs found
Multiple Factorizations of Bivariate Linear Partial Differential Operators
We study the case when a bivariate Linear Partial Differential Operator
(LPDO) of orders three or four has several different factorizations.
We prove that a third-order bivariate LPDO has a first-order left and right
factors such that their symbols are co-prime if and only if the operator has a
factorization into three factors, the left one of which is exactly the initial
left factor and the right one is exactly the initial right factor. We show that
the condition that the symbols of the initial left and right factors are
co-prime is essential, and that the analogous statement "as it is" is not true
for LPDOs of order four.
Then we consider completely reducible LPDOs, which are defined as an
intersection of principal ideals. Such operators may also be required to have
several different factorizations. Considering all possible cases, we ruled out
some of them from the consideration due to the first result of the paper. The
explicit formulae for the sufficient conditions for the complete reducibility
of an LPDO were found also
Thermodynamic phase transitions and shock singularities
We show that under rather general assumptions on the form of the entropy
function, the energy balance equation for a system in thermodynamic equilibrium
is equivalent to a set of nonlinear equations of hydrodynamic type. This set of
equations is integrable via the method of the characteristics and it provides
the equation of state for the gas. The shock wave catastrophe set identifies
the phase transition. A family of explicitly solvable models of
non-hydrodynamic type such as the classical plasma and the ideal Bose gas are
also discussed.Comment: revised version, 18 pages, 6 figure
The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies
The algebraic and Hamiltonian structures of the multicomponent dispersionless
Benney and Toda hierarchies are studied. This is achieved by using a modified
set of variables for which there is a symmetry between the basic fields. This
symmetry enables formulae normally given implicitly in terms of residues, such
as conserved charges and fluxes, to be calculated explicitly. As a corollary of
these results the equivalence of the Benney and Toda hierarchies is
established. It is further shown that such quantities may be expressed in terms
of generalized hypergeometric functions, the simplest example involving
Legendre polynomials. These results are then extended to systems derived from a
rational Lax function and a logarithmic function. Various reductions are also
studied.Comment: 29 pages, LaTe
Singular limit of Hele-Shaw flow and dispersive regularization of shock waves
We study a family of solutions to the Saffman-Taylor problem with zero
surface tension at a critical regime. In this regime, the interface develops a
thin singular finger. The flow of an isolated finger is given by the Whitham
equations for the KdV integrable hierarchy. We show that the flow describing
bubble break-off is identical to the Gurevich-Pitaevsky solution for
regularization of shock waves in dispersive media. The method provides a scheme
for the continuation of the flow through singularites.Comment: Some typos corrected, added journal referenc
Identification method based on Zadeh filter models
Mathematical modeling which provides the description of objects and proper organization of control operations in future is an integral stage in the automation of production. One of the approaches to build a mathematical model of an object is to represent nonlinear systems as combinations of inertial and nonlinear inertialess elements. The models thus obtained are called block-oriented. In this paper, we consider nonlinear dynamic objects represented as the models of the Zadeh filter class. In the process of the method development the identification equations were derived for the case when the test signal is a single sinusoid. Then the case of two sinusoids was considered. Such investigations allowed us to identify the patterns and describe the general case for several test components in the signal. The results of digital modeling using the sum of harmonic signals confirm the feasibility and validity of the proposed approach for identifying nonlinear models of the Zadeh filter class
Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation
We demonstrate how the Moutard transformation of two-dimensional Schrodinger
operators acts on the Faddeev eigenfunctions on the zero energy level and
present some explicitly computed examples of such eigenfunctions for smooth
fast decaying potentials of operators with non-trivial kernel and for deformed
potentials which correspond to blowing up solutions of the Novikov-Veselov
equation.Comment: 11 pages, final remarks are adde
Hyperdeterminants as integrable discrete systems
We give the basic definitions and some theoretical results about
hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability
(understood as 4d-consistency) of a nonlinear difference equation defined by
the 2x2x2-hyperdeterminant. This result gives rise to the following hypothesis:
the difference equations defined by hyperdeterminants of any size are
integrable.
We show that this hypothesis already fails in the case of the
2x2x2x2-hyperdeterminant.Comment: Standard LaTeX, 11 pages. v2: corrected a small misprint in the
abstrac
Differential reductions of the Kadomtsev-Petviashvili equation and associated higher dimensional nonlinear PDEs
We represent an algorithm allowing one to construct new classes of partially
integrable multidimensional nonlinear partial differential equations (PDEs)
starting with the special type of solutions to the (1+1)-dimensional hierarchy
of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the
B\"urgers hierarchy).
We derive examples of four-dimensional nonlinear matrix PDEs together with
they scalar and three-dimensional reductions. Variants of the
Kadomtsev-Petviashvili type and Korteweg-de Vries type equations are
represented among them. Our algorithm is based on the combination of two
Frobenius type reductions and special differential reduction imposed on the
matrix fields of integrable PDEs. It is shown that the derived four-dimensional
nonlinear PDEs admit arbitrary functions of two variables in their solution
spaces which clarifies the integrability degree of these PDEs.Comment: 20 pages, 1 fugur
Dressing method based on homogeneous Fredholm equation: quasilinear PDEs in multidimensions
In this paper we develop a dressing method for constructing and solving some
classes of matrix quasi-linear Partial Differential Equations (PDEs) in
arbitrary dimensions. This method is based on a homogeneous integral equation
with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to
systems of non-differential (algebraic or transcendental) equations for the
unknown fields. In the simplest examples, the above dressing scheme captures
matrix equations integrated by the characteristics method and nonlinear PDEs
associated with matrix Hopf-Cole transformations.Comment: 31 page
Dispersionful analogues of Benney's equations and -wave systems
We recall Krichever's construction of additional flows to Benney's hierarchy,
attached to poles at finite distance of the Lax operator. Then we construct a
``dispersionful'' analogue of this hierarchy, in which the role of poles at
finite distance is played by Miura fields. We connect this hierarchy with
-wave systems, and prove several facts about the latter (Lax representation,
Chern-Simons-type Lagrangian, connection with Liouville equation,
-functions).Comment: 12 pages, latex, no figure
- …