Matrix Sturm–Liouville equation with a Bessel-type singularity on a finite interval

The mKdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k-plane. This Riemann-Hilbert problem has explicit (x, t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of t...

Exponentially slow traveling waves on a finite interval for Burgers’ type equation

In this paper we study for small positive the slow motion of the solution for evolution equations of Burgers’ type with small diffusion, ut = uxx + f(u)ux , u(x, 0) = u0(x), u(±1, t) = ±1, (?) on the bounded spatial domain [−1, 1]; f is a smooth function satisfying f(1) > 0, f(−1) < 0 and ∫ 1 −1 f(t)dt = 0. The initial and boundary value problem (?) has a unique asymptotically stable equilibriu...

On the solving of matrix equation of Sylvester type

A solution of two problems related to the matrix equation of Sylvester type is given. In the first problem, the procedures for linear matrix inequalities are used to construct the solution of this equation. In the second problem, when a matrix is given which is not a solution of this equation, it is required to find such solution of the original equation, which most accurately approximates the ...

Finite Gorenstein Representation Type Implies Simple Singularity

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to...

A Convolution Equation on a Compact Interval