The interaction between two bright solitons in a dipolar Bose-Einstein condensate (BEC) has been investigated aiming at finding the regimes when they form a stable bound state, known as soliton molecule. To study soliton interactions in BEC we employed a method similar to that used in experimental investigation of the interaction between solitons in optical fibers. The idea consists in creating two solitons at some spatial separation from each other at initial time t0, and then measuring the distance between them at a later time t1 > t0. Depending on whether the distance between solitons has increased, decreased or remained unchanged, compared to its initial value at t0, we conclude that soliton interaction was repulsive, attractive or neutral, respectively. We propose an experimentally viable method for estimating the binding energy of a soliton molecule, based on its dissociation at critical soliton velocity. Our theoretical analysis is based on the variational approach, which appears to be quite accurate in describing the properties of soliton molecules in dipolar BEC, as reflected in good agreement between the analytical and numerical results.
Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix elements of the generalized Yukawa potential with complex screening parameter. This enabled us to treat analytically both the cosine and sine-like Yukawa potentials on equal footing and compute their bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we used a carefully designed complex scaling method to evaluate the resonance energies and compared our results satisfactorily with those obtained in the literature. The screened Coulomb potential is used in various areas of physics to model singular but short-range interactions [1]. In high energy physics, for example, it is used to model the interaction of hadrons in short range gauge theories where coupling is mediated by the exchange of a massive scalar meson [1,2]. In atomic and molecular physics, it represents a screened Coulomb potential due to the cloud of electronic charges around the nucleus, which could be treated in the Thomas-Fermi approximation that leads to [3] where µ is the screening parameter and A is the potential strength. This potential also describes the shielding effect of ions embedded in plasmas where it is called the DebyeHückel potential [4]. It has also been used to describe the interaction between charged particles in plasmas, solids and colloidal suspensions [5]. The exponential cosine screened Coulomb potential (ECSC) defined byhas been used to describe the long range interaction between an ionized impurity and an electron in a metal or semiconductor [6]. This oscillating potential was also used to describe the electron-positron interaction in a positronium atom in a solid [7]. This potential and the original Yukawa potential (1) can be lumped in a single complex Yukawa potential Where A, R µ and I µ are real positive parameters describing the strength of the potential and the real and imaginary parts of the screening parameter. The classical Yukawa potential (1) is obtained from (3) by choosing a real screening parameter, 0 I µ = , while the cosine like Yukawa potential (2) and the sine-like Yukawa potential are obtained by taking the real and imaginary parts of (3), respectively.
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