415,379 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
On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields
It was shown recently that Frobenius reduction of the matrix fields reveals
interesting relations among the nonlinear Partial Differential Equations (PDEs)
integrable by the Inverse Spectral Transform Method (-integrable PDEs),
linearizable by the
Hoph-Cole substitution (-integrable PDEs) and integrable by the method of
characteristics (-integrable PDEs). However, only two classes of
-integrable PDEs have been involved: soliton equations like Korteweg-de
Vries, Nonlinear Shr\"odinger, Kadomtsev-Petviashvili and Davey-Stewartson
equations, and GL(N,\CC) Self-dual type PDEs, like Yang-Mills equation. In
this paper we consider the simple five-dimensional nonlinear PDE from another
class of -integrable PDEs, namely, scalar nonlinear PDE which is
commutativity condition of the pair of vector fields. We show its origin from
the (1+1)-dimensional hierarchy of -integrable PDEs after certain
composition of Frobenius type and differential reductions imposed on the matrix
fields. Matrix generalization of the above scalar nonlinear PDE will be derived
as well.Comment: 14 pages, 1 figur
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
Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables
Using matrix identities, we construct explicit pseudo-exponential-type
solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two
variables and of nonlinear wave equations depending on three variables
Positive Definite Solutions of the Nonlinear Matrix Equation
This paper is concerned with the positive definite solutions to the matrix
equation where is the unknown and is
a given complex matrix. By introducing and studying a matrix operator on
complex matrices, it is shown that the existence of positive definite solutions
of this class of nonlinear matrix equations is equivalent to the existence of
positive definite solutions of the nonlinear matrix equation
which has been extensively studied in the
literature, where is a real matrix and is uniquely determined by It is
also shown that if the considered nonlinear matrix equation has a positive
definite solution, then it has the maximal and minimal solutions. Bounds of the
positive definite solutions are also established in terms of matrix .
Finally some sufficient conditions and necessary conditions for the existence
of positive definite solutions of the equations are also proposed
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