Soliton solutions and quasiperiodic solutions of modified Korteweg-de Vries type equations

A hierarchy of new nonlinear evolution equations which contains the modified Korteweg-de Vries equation is proposed. With the aid of the inverse scattering transformation, N-soliton solutions of the first two nonlinear evolution equations in this hierarchy are derived. Based on the theory of algebraic curve, the corresponding flows are straightened under the Abel-Jacobi coordinates. The meromorphic function phi and the hyperelliptic curve K-n are introduced by which quasiperiodic solutions of the first two nonlinear evolution equations are constructed according to the asymptotic properties and the algebrogeometric characters of phi and K-n. (C) 2010 American Institute of Physics. [doi :10.1063/1.3409345]