Soliton and traveling wave solutions in coupled one-dimensional condensates

Ultracold condensates provide a unique platform for exploring soliton physics. Motivated by the recent experiments realizing the sine-Gordon model in a split one-dimensional (1D) BEC, we demonstrate that this system naturally supports various density and phase solitons. We explore the physics using the bosonization technique, in which the phase and density are conjugate pairs, and determine its effective Language equation and the associated equation of motion. We show that in the presence of asymmetry between the two condensates, new solutions beyond those in the sine-Gordon model emerge. We calculate the traveling wave solutions and soliton solutions in this model and determine their corresponding energy densities analytically. Finally, we discuss the relevance of these solutions to the experiments and discuss their observations. This theory does not rely on the mechanism of quasi-particle excitation, which yields the Lee-Huang-Yang correction in higher dimensions, and is thus much more suitable to describe the physics in 1D systems. Since the physical models have already been realized in experiments, this work opens a new frontier for the realization of various soliton and periodic solutions using two coupled condensates.