Integrable structures of a Bose-Einstein condensate model: From asymptotic reduction to exact solutions and conservation laws

This paper studies a nonlinear Schrödinger-type model for Bose-Einstein condensates with a fixed $s$-wave scattering length. Under a specific rescaling, the dimensionless Gross-Pitaevskii equation takes the form $i{\mathrm{\Psi }}_t=-{\mathrm{\nabla }}^2\mathrm{\Psi }-2|\mathrm{\Psi }{|}^2\mathrm{\Psi }.$ The model is rewritten in hydrodynamic form via a density-phase decomposition, and in the long-wave limit, it is asymptotically reduced to the classical Korteweg-de Vries (KdV) equation. For this reduced KdV equation, we present its Lax pair, Darboux transformation, and Bäcklund transformation, and obtain multi-soliton and positon solutions by means of the Hirota bilinear method. Furthermore, by coupling the tau functions of a single soliton and a first-order positon via the Wronskian determinant, we construct a mixed interaction solution, whose explicit expression contains nonlinear cross-coupling terms which characterize the collision dynamics between the two types of waves. An asymptotic expansion of the spectral problem yields infinitely many conservation laws, and we prove that all orders of global conserved quantities for the soliton-positon composite wave satisfy an exact superposition relation. These results demonstrate that the reduced KdV equation derived from the Bose-Einstein condensate model possesses a complete integrable structure, thus providing a theoretical reference for understanding nonlinear wave propagation in quantum fluids under the KdV approximation.