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

The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions

“…3 (b) and (c) above, we are not able to derive nontrivial initial solutions using the same fixed point iteration method of [22,Algorithm 1]. We resolve the problem by following the solution branch of (a) to a particular value of λ at which we expect the solution to exist for (b) or (c), then change the value of a to the assigned value for (b) or (c) to compute for a nontrivial initial solution.…”

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

“…If the sign of the smallest eigenvalues changes, a bifurcation point between two computed successive solution points exists. We then refine further to find the bifurcation point accurately by a local search, as suggested in [21,24]. In a detected bifurcation point, the corresponding eigenvector and its opposite (with an additional negative sign) are used as the tangent vectors to the solution branches that depart from the bifurcation point.…”

“…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.…”

“…In particular, we consider a condensate without the trapping potential (i.e., e V ext ¼ 0). To explore the effect of g s , a so-called ''parameter-switching technique'' [21] is applied to the PACM. We first track the solution curve C ð1Þ 3 ¼ ðx; sÞjFðx; sÞ ¼ 0; with g n ¼ Às; g s ¼ À0:2s; for 10 À5 6 s 6 1…”

“…The multi-component BECs with frozen spin degrees of freedom based on the coupled nonlinear Schrödinger equations (CNSE) have been investigated analytically and numerically [8,9,13,21]. Recently, Cao et al [17] prove the existence of the ground state for the spin-1 BEC in some cases.…”