152 research outputs found
Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions
Hamiltonian systems of hydrodynamic type occur in a wide range of
applications including fluid dynamics, the Whitham averaging procedure and the
theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the
integrability of such systems by the generalised hodograph transform implies
that integrable Hamiltonians depend on a certain number of arbitrary functions
of two variables. On the contrary, in 2+1 dimensions the requirement of the
integrability by the method of hydrodynamic reductions, which is a natural
analogue of the generalised hodograph transform in higher dimensions, leads to
finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we
classify integrable two-component Hamiltonian systems of hydrodynamic type for
all existing classes of differential-geometric Poisson brackets in 2D,
establishing a parametrisation of integrable Hamiltonians via
elliptic/hypergeometric functions. Our approach is based on the Godunov-type
representation of Hamiltonian systems, and utilises a novel construction of
Godunov's systems in terms of generalised hypergeometric functions.Comment: Latex, 34 page
A multidimensionally consistent version of Hirota's discrete KdV equation
A multidimensionally consistent generalisation of Hirota's discrete KdV
equation is proposed, it is a quad equation defined by a polynomial that is
quadratic in each variable. Soliton solutions and interpretation of the model
as superposition principle are given. It is discussed how an important property
of the defining polynomial, a factorisation of discriminants, appears also in
the few other known discrete integrable multi-quadratic models.Comment: 11 pages, 2 figure
Lattice and q-difference Darboux-Zakharov-Manakov systems via -dressing method
A general scheme is proposed for introduction of lattice and q-difference
variables to integrable hierarchies in frame of -dressing
method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov
systems of equations are derived. Darboux, B\"acklund and Combescure
transformations and exact solutions for these systems are studied.Comment: 8 pages, LaTeX, to be published in J Phys A, Letters
Invariant description of solutions of hydrodynamic type systems in hodograph space: hydrodynamic surfaces
Hydrodynamic surfaces are solutions of hydrodynamic type systems viewed as
non-parametrized submanifolds of the hodograph space. We propose an invariant
differential-geometric characterization of hydrodynamic surfaces by expressing
the curvature form of the characteristic web in terms of the reciprocal
invariants.Comment: 12 page
Flat bidifferential ideals and semihamiltonian PDEs
In this paper we consider a class of semihamiltonian systems characterized by
the existence of a special conservation law.
The density and the current of this conservation law satisfy a second order
system of PDEs which has a natural interpretation in the theory of flat
bifferential ideals. The class of systems we consider contains important
well-known examples of semihamiltonian systems. Other examples, like genus 1
Whitham modulation equations for KdV, are related to this class by a
reciprocal trasformation.Comment: 18 pages. v5: formula (36) corrected; minor change
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
Hydrodynamic reductions of the heavenly equation
We demonstrate that Pleba\'nski's first heavenly equation decouples in
infinitely many ways into a triple of commuting (1+1)-dimensional systems of
hydrodynamic type which satisfy the Egorov property. Solving these systems by
the generalized hodograph method, one can construct exact solutions of the
heavenly equation parametrized by arbitrary functions of a single variable. We
discuss explicit examples of hydrodynamic reductions associated with the
equations of one-dimensional nonlinear elasticity, linearly degenerate systems
and the equations of associativity.Comment: 14 page
S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies
We introduce an S-function formulation for the recently found r-th
dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also
a connection of these -functions with the Orlov functions of the
hierarchies. Then, we discuss a reduction scheme for the hierarchies that
together with the -function formulation leads to hodograph systems for the
associated solutions. We consider also the connection of these reductions with
those of the dispersionless KP hierarchy and with hydrodynamic type systems. In
particular, for the 1-component and 2-component reduction we derive, for both
hierarchies, ample sets of examples of explicit solutions.Comment: 35 pages, uses AMS-Latex, Hyperref, Geometry, Array and Babel
package
Critical Review of Theoretical Models for Anomalous Effects (Cold Fusion) in Deuterated Metals
We briefly summarize the reported anomalous effects in deuterated metals at
ambient temperature, commonly known as "Cold Fusion" (CF), with an emphasis on
important experiments as well as the theoretical basis for the opposition to
interpreting them as cold fusion. Then we critically examine more than 25
theoretical models for CF, including unusual nuclear and exotic chemical
hypotheses. We conclude that they do not explain the data.Comment: 51 pages, 4 Figure
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