Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDeterminant structure for tau-function of holonomic deformation of linear differential equations
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In our previous works, a relationship between Hermites two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study tau-functions associated with holonomic deformations of linear differential equations by using Hermites two approximation problems. As a result, we present a determinant formula for the ratio of tau-functions (tau-quotient).
rate research
Read More
Picone-type identities are established for half-linear ODEs of fourth order (one-dimensional p-biLaplacian). It is shown that in the linear case they reduce to the known identities for fourth order linear ODEs. Picone-type identity known for two half-linear second-order equations is also generalised to set of equations greater than two.
We consider the differential equation begin{align}label{ab} -y(x)+q(x)y(x)=f(x), quad x in mathbb R, end{align} where $f in L_{p}(mathbb R)$, $pin [1,infty)$, and $0leq q in L_{1}^{rm loc}(mathbb R)$, $intlimits_{-infty}^{0}q(t),dt=intlimits_{0}^{infty}q(t),dt=infty,$ begin{align*} q_{0}(a)=inf_{xin mathbb R}int_{x-a}^{x+a}q(t),dt=0 quad{rm for ~ any }quad ain (0,infty). end{align*} Under these conditions, the equation ({rm ref{ab}}) is not correctly solvable in $L_{p}(mathbb R)$ for any $p in [1, infty) $. Let $q^{*}(x)$ be the Otelbaev-type average of the function $q(t), tin mathbb{R}$, at the point $t=x$; $theta(x)$ be a continuous positive function for $x in mathbb R$, and begin{align*} L_{p,theta }(mathbb R) = {fin L_{p}^{rm loc}(mathbb R):, int_{-infty}^{infty}|theta(x)f(x)|^{p},dx<infty }, end{align*} begin{align*} |f|_{L_{p,theta}(mathbb R)}=left(int_{-infty}^{infty}|theta(x)f(x)|^{p},dxright)^{1/p} end{align*} We show that if there exists a constant $cin [1, infty)$, such that the inequality $$c^{-1}q^{*}(x)leq theta(x)leq cq^{*}(x)$$ holds for all $x in mathbb{R}$, then under some additional conditions for $q$ the pair of spaces ${L_{p, theta}(mathbb R); L_{p}(mathbb R)}$ is admissible for the equation ({rm ref{ab}}).
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.
Admissible point transformations of classes of $r$th order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first and second Arnold forms) are exhaustively described. Using these results, the group classification of such equations is revisited within the algebraic approach in three different ways.
Four 4-dimensional Painleve-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painleve system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painleve-type equations. If we only consider Painleve-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painleve-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are neatly written using Hamiltonians of the classical Painleve equations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
