Solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature

2025/06/08 by Kazuyuki Yagasaki, Yagasaki, Kazuyuki · 1 citation
Physics and Astronomy · Mathematics · #Quantum Mechanics and Non-Hermitian Physics #Nonlinear Waves and Solitons #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2506.08046

Abstract

We study the solvability of the Korteweg-de Vries equation under meromorphic initial conditions by quadrature when the inverse scattering transform (IST) is applied. It is a key to solve the Schrödinger equation appearing in the Lax pair in application of the IST. We show that the Schrödinger equation is always integrable in the sense of differential Galois theory, i.e., solvable by quadrature, if and only if the meromporphic potential is reflectionless, under the condition that the potential is absolutely integrable on ℝ∖(-R₀,R₀) for some R₀>0.This statement was previously proved to be true by the author for a limited class of potentials. We also show that the Schrödinger equation is not integrable in this sense for rational potentials that decay at infinity but do not satisfy the weak condition.

Citations

• Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature
• Integrability of the Zakharov-Shabat systems by quadrature
• Galoisian approach for a Sturm-Liouville problem on the infinite\n interval
• Galoisian Approach to integrability of Schrödinger Equation
• Introduction to the Galois Theory of Linear Differential Equations
• Solving the Korteweg-de Vries Equation by Its Bilinear Form: Wronskian Solutions

Cited by

• Solvability of the Zakharov-Shabat systems with meromorphic potentials by quadrature

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