Riemann-Hilbert approach to the algebro-geometric solution of the modified Camassa-Holm equation with linear dispersion term

This paper aims at providing an exact algebro-geometric solution of the modified Camassa-Holm (mCH) equation derived from hyperelliptic curves in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 left-parenthesis p plus q right-parenthesis minus 1"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">4(p+q)-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> genus. To achieve this goal, we construct the Riemann-Hilbert problems cosponsoring to the mCH equation, which can be solved exactly by the Baker-Akhiezer function. Then the precise expression of the algebro-geometric solution of the mCH equation can be obtained through reconstructed formula.