D. E. Pelinovsky and R. Weikard
Bright and dark breathers on an elliptic wave in the defocusing mKdV equation
Abstract:
Breathers on an elliptic wave background consist of nonlinear superpositions of a soliton and a periodic wave,
both traveling with different wave speeds and interacting periodically in the space-time.
For the defocusing modified Korteweg-de Vries (mKdV) equation, the construction of general breathers
has been an open problem since the elliptic wave is related to the elliptic degeneration of
the hyperelliptic solutions of genus two. We have found the new representation of eigenfunctions
of the Lax operator associated with the elliptic wave, which enables us to solve this open problem
and to construct two families of breathers with bright (elevation) and dark (depression) profiles.
Keywords:
modified Korteweg-de Vries equation, elliptic traveling waves, bright breathers, dark breathers,
Darboux transformation, Miura transformation, Jacobi's theta functions, Lame equation.