Rotation number and eigenvalues of two-component modified Camassa-Holm equations

In this paper, we study the spectral problem (������1) (- 1 ) 2 ������������ 1 ) (������1 = 2 ������2 -21 ������������ 1 ������2 ������ 2 for the two -component modified Camassa-Holm equation, where ������, ������are two potentials. By introducing the rotation number ������(������) and studying its properties, we prove that for any integer ������, { } the periodic or anti -periodic eigenvalues are the endpoints of the interval ������ is an element of R : ������(������) = - ������ . 2 Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space ������1[0, ������]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the ������1 norms of potentials are given.