2017/08/01 by Jiao Wei, Xianguo Geng, Xin Zeng · 1 citation
Computer Science · Physics and Astronomy · Mathematics · #Polynomial and algebraic computation #Nonlinear Waves and Solitons #Mathematical Dynamics and Fractals
paper · pdf · doi:10.1090/tran/7349
Starting with a discrete <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 times 3"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo> × </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3× 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix spectral problem, the hierarchy of Bogoyavlensky lattices which are pure differential-difference equations are derived with the aid of the Lenard recursion equations and the stationary discrete zero-curvature equation. By using the characteristic polynomial of Lax matrix for the hierarchy of stationary Bogoyavlensky lattices, we introduce a trigonal curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K Subscript m minus 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal Km-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of arithmetic genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m minus 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a basis of holomorphic differentials on it, from which we construct the Riemann theta function of the trigonal curve, the related Baker–Akhiezer function, and an algebraic function carrying the data of the divisor. Based on the theory of trigonal curves, the Riemann theta function representations of the Baker–Akhiezer function, the meromorphic function, and in particular, that of solutions of the hierarchy of Bogoyavlensky lattices are obtained.