Effective Mass in Bose-Einstein Condensation in the Bound State and   Phonon Propagation in the Unbound States
  The dark and bright solitons in different systems are already known in Klein-Gordon lattice. Instead of an external driving force, if the intrinsic field is only considered, then the modal dynamics for small oscillations could be characterized by the bound state in a limited range of frequency, revealed via associated Legendre polynomial. Bose Einstein condensation takes place around bosonic particles having different wave functions within the bound states in the temperature region T = 0 to Tc having implication for the effective mass of the system. The pairing and interplay between the dark and bright solitons also occur with their effect on the condensation. This effective mass is calculated via statistical mechanics route by two-part partition function that also gives an indication for the transition temperature. The disappearance of the bound state after a critical frequency, or equivalently, after a critical temperature, gives rise to quasi-particles or phonons in the unbound states that propagate through the domains. 
Asis. K. Bandyopadhyay, Atrayee Bhattacharya, Kamal Choudhary, and   Santanu Das
cond-mat.quant-gas