Dear Kuldeep Suthar,

      Could you please give any reference for understanding the basic idea of vortices and the reference for solitonic vortex in BEC?

Thanking You

Sincerely Yours

J B Sudharsan

Dear Kuldeep,

Soliton, soliton vortex and vortex all are topological defects in which atoms have collective motions.  The difference can be understood by using some imagination about the atomic arrangement in a many body system.   Say for example in a Bose Einstein condensate of trapped dilute gases where this distinction becomes very clear if the trapping potential quenches the transverse motion of atoms and the condensate  takes the shape of a cigar.   Since the low energy excitations which are the only possibilities in such a system at its low temperature states, atoms can be seen to be moving collective either along the length of the cigar or in opposite direction.  However , this would represent the ground state motion, the next excited  state would be one in which half of the cigar can be seen to have atoms moving in one direction and remaining half would have atoms moving in opposite direction.  This is what people call a solition motion.  On the other hand there is another possibility in which atoms move around the axis of the cigar collectively in order of their locations.  This is what they call it soliton vortex.   Soliton vortex is also a quantized vortex.  However, it differs from a quantized vortex in 2-D and 3-D system superfluids as in the former case it represents a collective motion of atoms in superfluid surface (a planar arrangement of atoms), in the latter case it is a vortex which need not be linear; it can have many bindings along its large length which can be visualized to be curve that you can draw in 3-D space.   However, please remember that solitons and vortices do exist in classical liquids too but there the atomic motions are not coherent.  

Studies of vortices, and their relations with 1D DARK solitons are provided in my book on nonlinear light patterns. The book can be downloaded from my account in reseach gate. These vortices were studied essentially in context of lasers there, but in dynanical sense they are identic to those in BECs. Good luck with vortices! Best. K.S. 

Thanks Prof. Jain, Nice explanation.  What would be the phase structure around the  solitonic vortex, as it is well defined in the case of the soliton or vortex ?

For solitary waves, there are two types. Bright solitons can exist in BECs with attractive interactions, while dark solitons are topological excitation in repulsive BECs.  Bright solitons are self-focusing, nondispersive, particlelike solitary waves (see PRA.85.013627 for detailed description of one dimensional bright solitons in condensates with and without a trap).  They have uniform phase distribution. A dark soliton is an envelope soliton that has the form of a density dip with a phase jump across its density minimum (see review in J. Phys. A: Math. Theor. 43 (2010) 213001).  Vortices are also topological excitations in condensates. The quantization of circulation of the fluid leads to quantized vortex states which are characterized by the phase circulation about the vortex core being $2s\pi$ (Phys. Rev. Lett. 81, 5477, Phys. Rev. A 64, 031601(R) ), where $s$, the winding number or vortex charge, is a nonzero integer. This phase singularity is independent of atomic and trap parameters. Vortices with opposite charge (opposite phase circulation, i.e. clockwise and anticlockwise) are called vortex and antivortex. If the winding number satisfies the relation $|s|>1$, vortices are not stable when perturbed. They will dissociate into singly charged vortics ($s=\pm 1$). 

Svortices are vortices confined to essentially one-dimensional dynamics, which obey a similar phase-offset–velocity relationship as solitons. Marking the transition between solitons and vortices, svortices are a distinct class of symmetry-breaking stationary and uniformly rotating excited solutions of the 2D and 3D Gross-Pitaevskii equation, , which have properties of both vortices and solitons (J. Phys. B: At. Mol. Opt. Phys. 34 L113, New J. Phys. 15 113028).

the phase structure around the solitonic vortex, too is well defined and is not different from a vortex.

Perhaps the following article would be highly useful for all people interested in this topic.

Frédéric Chevy, Physics 7, 82 (2014) – Published August 4, 2014

In what follows from

Frédéric Chevy, Physics 7, 82 (2014) – Published August 4, 2014

it is evident that the experimental observation of solitons,  solitonic vortices in BEC state in trapped dilute gases unequivocally proves that atoms in this state do not move randomly and they do not have relative motion;  they rather move coherently in order of their position.   This obviously contradict conventional belief that atoms in BEC state of trapped dilute gases have large fraction of atoms with p = 0 condensate and remaining ones (having equally large number) with non-zero p.  This implies that atoms have relative motions for which they have mutual collisions leading to random change the phases of different atoms which again is inconsistent with their coherent motion.

This is more a remark, or a caveat,  than an answer: a "soliton", strictly speaking, is a special solution with precise methematical properties (e.g., the supporting equation should be integrable). Thus, in real systems where the underlying equation is obviously not exactly known, one can just speculate the "soliton" nature of the solution. Also, in general it is not true that solitons are "topological defects" -- that properly applies only to the subclass of "topological solitons". This should be carefully considered in discussing the topic with mathematically oriented people, while in physics the term soliton is used more freely.