3,250 research outputs found
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
Some properties of solutions to weakly hypoelliptic equations
A linear different operator L is called weakly hypoelliptic if any local
solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients
may be matrices, not necessarily of square size. This is a huge class of
important operators which cover all elliptic, overdetermined elliptic,
subelliptic and parabolic equations.
We extend several classical theorems from complex analysis to solutions of
any weakly hypoelliptic equation: the Montel theorem providing convergent
subsequences, the Vitali theorem ensuring convergence of a given sequence and
Riemann's first removable singularity theorem. In the case of constant
coefficients we show that Liouville's theorem holds, any bounded solution must
be constant and any L^p-solution must vanish.Comment: published version (up to cosmetic issues
Geometry of Darboux-Manakov-Zakharov systems and its application
The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov
systems of semilinear partial differential equations \label{GDMZabstract}
\frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial
u}{\partial x_l}\Big), 1\leq i<j\leq n, k,l\in\{1,...,n\} for a real-valued
function are studied with particular reference to the linear
systems in this equation class.
System (\ref{GDMZabstract}) will not generally be involutive in the sense of
Cartan: its coefficients will be constrained by complicated nonlinear
integrability conditions. We derive geometric tools for explicitly constructing
involutive systems of the form (\ref{GDMZabstract}), essentially solving the
integrability conditions. Specializing to the linear case provides us with a
novel way of viewing and solving the multi-dimensional -wave resonant
interaction system and its modified version as well as constructing new
examples of semi-Hamiltonian systems of hydrodynamic type. The general theory
is illustrated by a study of these applications
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
Numerical Schubert calculus
We develop numerical homotopy algorithms for solving systems of polynomial
equations arising from the classical Schubert calculus. These homotopies are
optimal in that generically no paths diverge. For problems defined by
hypersurface Schubert conditions we give two algorithms based on extrinsic
deformations of the Grassmannian: one is derived from a Gr\"obner basis for the
Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its
projective coordinate ring. The more general case of special Schubert
conditions is solved by delicate intrinsic deformations, called Pieri
homotopies, which first arose in the study of enumerative geometry over the
real numbers. Computational results are presented and applications to control
theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st
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