A hyperplane-constrained continuation method for near singularity in coupled nonlinear Schrödinger equations

Exploring ground states and excited states of spin-1 Bose–Einstein condensates by continuation methods

“…that has an additional constraint as suggested in [22]. This constraint forces the next solution point u i+1 lies on the hyperplane passing u ð1Þ iþ1 and being orthogonal to the vector _ u i .…”

“…We first briefly introduce the framework of PACM and refer to [2,6,21,22,24] for details. Then we discuss how to determine initial points that can lead to ground state solutions of spin-1 BEC.…”

A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

“…The theorem gives conditions under which a point on the solution curve is a bifurcation point. The theoretical and numerical details for detecting bifurcation points of the solution curve C and for tracing the bifurcation branches can be found in [8,18].…”

“…In the following, we shall focus on problem (18). Taking gradient of e EðuÞ, we have r e EðuÞ ¼ Au À ðu s> u s Þ À1=2 u t :…”

“…Simple straightforward numerical methods cannot lead to non-radially symmetric solutions. For example, only radially symmetric solutions are found for small b ij 's if we simply follow the solution curve C b ¼ fðx > ; bÞ > j Gðx; bÞ ¼ 0 is given inð38Þ with d 12 ¼ d 13 ¼ 1 and d 23 ¼ À1; for b 2 R þ g: by starting from b ¼ 0 [18]. Now, we describe how we can find the non-radially symmetric solutions for the case of one repulsive and two attractive interactions in the 3-component DNLS Eqs.…”