Solitons in nonlinear systems and eigen-states in quantum wells

Abstract

We study the relations between solitons of nonlinear Schrödinger equation described systems and eigen-states of linear Schrödinger equation with some quantum wells. Many different nondegenerated solitons are re-derived from the eigen-states in the quantum wells. We show that the vector solitons for coupled system with attractive interactions correspond to the identical eigenstates with the ones of coupled systems with repulsive interactions. The energy eigenvalues of them seem to be different, but they can be reduced to identical ones in the same quantum wells. The non-degenerated solitons for multi-component systems can be used to construct much abundant degenerated solitons in more components coupled cases. On the other hand, we demonstrate soliton solutions in nonlinear systems can be also used to solve the eigen-problems of quantum wells. As an example, we present eigenvalue and eigen-state in a complicated quantum well for which the Hamiltonian belongs to the non-Hermitian Hamiltonian having Parity-Time symmetry. We further present the ground state and the first exited state in an asymmetric quantum double-well from asymmetric solitons. Based on these results, we expect that many nonlinear physical systems can be used to observe the quantum states evolution of quantum wells, such as water wave tank, nonlinear fiber, Bose-Einstein condensate, and even plasma, although some of them are classical physical systems. These relations provide another different way to understand the stability of solitons in nonlinear Schrödinger equation described systems, in contrast to the balance between dispersion and nonlinearity.

Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates

“…Secondly, we investigate the interaction between two NDBSSs by performing forth-fold DT with spectral parameters λ 1 = a 1 + ib 1 , λ 2 = a 1 + ib 2 (generate one NDBSS), λ 3 = a 2 + ib 3 , and λ 4 = a 2 + ib 4 (generate the other NDBSS). We exhibit the dynamical evolution of them in the right panel of Fig.3, based on the two double-hump solitons solution (11). (b1) and (b2) correspond to component q 1 and component q 2 respectively.…”

“…The effective quantum well for this three-component case is −2|q 1 | 2 − 2|q 2 | 2 − 2|q 3 | 2 , and it is a triple-well form. Based on the the correspondence between solitons and eigen-states in quantum wells [11,31], one can know that the triple-hump bright soliton with no node in q 3 component is ground state (see green dashed line in Fig. 5), the triple-hump bright soliton with one node in q 2 component is the first-excited state (see blue dotted-dashed line in Fig.…”

“…Many different vector solitons have been obtained in the two-component coupled BEC systems, such as the bright-bright soliton [2,3], the bright-dark soliton [4], the dark-antidark soliton [5], the dark-dark soliton [6,7], and the dark-bright soliton [8,9]. The soliton states can be related with eigen-states in quantum well [10,11]. From the general properties of eigen-states in one-dimensional quantum wells, one can know that fundamental bright soliton corresponds to ground state and dark soliton is first-excited state in the effective quantum wells.…”

“…With the aid of Darboux transformation (DT) [23][24][25][26] or Hirota method [27,28], many different vector solitons have been obtained in the two-component coupled BEC systems, such as the bright-bright soliton [2,29,30], the bright-dark soliton [4], the dark-dark soliton [6,7], and the dark-bright soliton [8,9]. From the relations between soliton and eigen-states in quantum well [10,11], we can know that bright-bright soliton and dark-dark soliton are degenerate solitons (more than one component admits the same spatial mode), bright-dark soliton and darkbright soliton are non-degenerate soliton states. However, the dark soliton state is a free state, and bright soliton usually admits no nodes [2,29,30].…”

“…When b 1 < b 2 , soliton in q 2 component admits no node, and the one in q 1 component always has one node, and vice versa. From the general properties of bound states in one-dimensional potential [31], we know that the eigen-state with one node corresponds to the first-excited state in the self-induced effective quantum well [11]. Therefore, the bound state solitons in the two components correspond to the ground state and the first-excited state in the effective quantum well respectively.…”