Stability of multi-parameter solitons: Asymptotic approach

General asymptotic approach to the stability problem of multi-parameter solitons in Hamiltonian systems $i\partial E_n/\partial z=\delta H/\delta E_n^*$ has been developed. It has been shown that asymptotic study of the soliton stability can be reduced to the calculation of a certain sequence of the determinants, where the famous determinant of the matrix consisting from the derivatives of the system invariants with respect to the soliton parameters is just the first in the series. The presented approach gives first analytical criterion for the oscillatory instability and also predicts novel stationary instability. Higher order approximations allow to calculate corresponding eigenvalues with arbitrary accuracy.Comment: to appear in Physica

Instabilities of vortices in a binary mixture of trapped Bose-Einstein condensates: Role of excitations with positive and negative energies

Correspondence between frequency and energy spectra and biorthogonality conditions for the excitations of Bose-Einstein condensates described by Gross-Pitaevskii model have been derived selfconsistently revealing several novel aspects originating in nonselfadjoitness of the Bogoliubov operator. It has been demonstrated that frequency resonances of the excitations with positive and negative energies can lead to their mutual annihilation and appearance of the collective modes with complex frequencies and zero energies. Conditions for the avoided crossing of energy levels have also been discussed. General theory has been verified both numerically and analytically in the weak interaction limit considering an example of vortices in a binary mixture of condensates. Growth of excitations with complex frequencies leads to spiraling of the unit and double vortices out of the condensate center to its periphery and to splitting of the double and higher order vortices to the unit ones.Comment: Physical Review A (January, 2001). Version 3 differs dramatically from v1 and v2 and contains significant amount of qualitatively new materia