On exact finite-gap solutions of the negative-order Korteweg–de Vries equation

G. U. Urazboev^a, M. M. Khasanov<sup>b

a</sup> Urgench State University named after Abu RayhonBeruni, Urgench, Uzbekistan
^b Khorezm Branch of the Romanovsky Institute of Mathematics, Uzbekistan Academy of Sciences, Urgench, Uzbekistan

Abstract: We show that the negative-order Korteweg–de Vries equation can be integrated using the inverse spectral problem method. We find the evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the finite-gap solution of the negative-order Korteweg–de Vries equation. The obtained results allow the inverse problem method to be applied to solve the negative-order Korteweg–de Vries equation in the class of periodic functions. We prove important implications regarding the analyticity and the spatial period of the finite-gap solution. We show that the solution constructed by the Dubrovin system of equations and the first trace formula satisfies the negative-order Korteweg–de Vries equation.

Keywords: negative-order Korteweg–de Vries equation, inverse spectral problem, Dubrovin system of equations, finite-gap solution, trace formulas.

Received: 19.06.2025
Revised: 15.08.2025

DOI: 10.4213/tmf11030

 English version: Theoretical and Mathematical Physics, 2026, 226:1, 87–96

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