Painlevé property, Lax pair and Darboux transformation of the variable-coefficient modified Kortweg-de Vries model in fluid-filled elastic tubes

New exact solutions for the variable coefficient modified KdV equation using direct reduction method

“…When h. / D 0, Equation (2) reduces to the well-known mKdV equation [5,6], whose bell-shaped solitary wave solution is obtained and shows that the wave speed is propagation to the square of the amplitude [7]. In [8], the Painlevé property of the vc-mKdV equation, the Lax pair, and the Darboux transformation are constructed, and soliton solution is obtained. On the other hand, when k. / ¤ 0, [1,2] also obtain the solitary wave solution of Equation (2).…”