On the absence of logarithmic singularities in the solutions of Lamé-type equations

E R Babich,I P Martynov

doi:10.29235/1561-2430-2021-57-4-428-434

Abstract

The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations. These equations will be called Lamé-type equations. The question arises under what conditions the general solution of Lamé-type equations contains no logarithms. For this purpose, in the present paper, the solutions of Lamé-type equations are investigated and the conditions are found that make it possible to judge the presence or absence of logarithmic singularities in the solutions of the equations under study. An example of an equation with an irregular singularity having a solution with an logarithmic singularity is given, since the equation, defining it, has a multiple root.

Highlights

The object of this research is linear differential equations of the second order with regular singularities
We extend the concept of a regular singularity to linear partial differential equations
The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations

Summary

Introduction

The object of this research is linear differential equations of the second order with regular singularities. Так как решение уравнения (5) имеет полярную особенность не выше второго порядка, то можем говорить о том, что уравнение (4) – уравнение с регулярной особенностью, аналогично тому, как применяется понятие регулярной особой точки в аналитической теории обыкновенных дифференциальных уравнений. При ν = 1 общее решение уравнения (1), где р удовлетворяет уравнению (2), не содержит логарифма.

Results

Conclusion