Search results for "Singular solution"

Numerical study of the long wavelength limit of the Toda lattice

2014

We present the first detailed numerical study of the Toda equations in $2+1$ dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the formal dispersionless limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up of the solution in the hyperbolic case is similar to the shock formation in the Hopf equation, a $1+1$ dimensional singularity. In the elliptic case, it is found that the break-up is given by a cusp as for the semiclassical system of the focusing nonlinear Schr\"odinger equation in $1+1$ dimensions. The full Toda system is then studie…

A Singular Multi-Grid Iteration Method for Bifurcation Problems

1984

We propose an efficient technique for the numerical computation of bifurcating branches of solutions of large sparse systems of nonlinear, parameter-dependent equations. The algorithm consists of a nested iteration procedure employing a multi-grid method for singular problems. The basic iteration scheme is related to the Lyapounov-Schmidt method and is widely used for proving the existence of bifurcating solutions. We present numerical examples which confirm the efficiency of the algorithm.

On a singular boundary value problem for a second order ordinary differential equation

2000

Singular integrals, analytic capacity and rectifiability

1997

In this survey we study some interplay between classical complex analysis (removable sets for bounded analytic functions), harmonic analysis (singular integrals), and geometric measure theory (rectifiability).

Singular distributed parameter systems

1993

The paper deals with the distributed parameter systems described by coupled partial differential equations with singular matrix coefficients. Initial-boundary-value problems are considered in the light of both singular 1d systems theory and the Fourier approach to distributed parameter systems. The method presented in this paper gives the possibility of determining acceptable initial-boundary conditions. An illustrative example is given.

Cosmological solutions in theD=5 Einstein-Maxwell theory coupled to matter

1991

We study the Einstein-Maxwell theory in five dimensions coupled to matter in two distinct ways. In the first we reduce the Lagrangian to an effective four-dimension one and then we couple it to matter; in the second, we introduce matter directly in the original five-dimensional theory. In both cases we use a non trivial configuration for the Maxwell potential. We find non singular solutions which present a repulsive gravitational phase. When this phase is absent, the initial singularity is unavoidable.

Stability of Hamiltonian Systems of Two Degrees of Freedom and of Formally Conservative Mappings Near a Singular Point

1985

We restrict ourselves to the stability problems considered in our lecture because the length of this paper is limited. In contrast to the lecture, however, we consider here not only area preserving mappings but a more general class of mappings.

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

2001

Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

New solvability conditions for the Neumann problem for ordinary singular differential equations

2000

Higher order matrix differential equations with singular coefficient matrices

2015

In this article, the class of higher order linear matrix differential equations with constant coefficient matrices and stochastic process terms is studied. The coefficient of the highest order is considered to be singular; thus, rendering the response determination of such systems in a straightforward manner a difficult task. In this regard, the notion of the generalized inverse of a singular matrix is used for determining response statistics. Further, an application relevant to engineering dynamics problems is included.