Daisuke A. Takahashi, Shunji Tsuchiya, Ryosuke Yoshii, Muneto Nitta
The chiral Gross-Neveu model or equivalently the linearized Bogoliubov-de Gennes equation has been mapped to the nonlinear Schr\"odinger (NLS) hierarchy in the Ablowitz-Kaup-Newell-Segur formalism by Correa, Dunne and Plyushchay. We derive the general expression for exact fermionic solutions for all gap functions in the arbitrary order of the NLS hierarchy. We find that the energy spectrum of the n-th NLS hierarchy consists of n bands for localized states and two continuums for scattering states and 2n+2 band edges. As an illustration of our formalism, we present the three-kink solution with three fermion bound states, one of which is a zero-energy mode and the rests of which are finite-energy modes. The zero-mode wave function has the peaks around the outer two kinks if the three kinks are symmetrically placed, while the zero mode is localized on it if one of the outer kinks is separated.
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http://arxiv.org/abs/1205.3299
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