Pseudo-Arclength Continuation Algorithms for Binary Rydberg-Dressed Bose-Einstein Condensates

2016 ◽  
Vol 19 (4) ◽  
pp. 1067-1093 ◽  
Author(s):  
Sirilak Sriburadet ◽  
Y.-S. Wang ◽  
C.-S. Chien ◽  
Y. Shih
Keyword(s):  
Ground State ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
State Solution ◽  
Divide And Conquer ◽  
Continuation Methods ◽  
Solution Curve ◽  
Bose Einstein ◽  
Curve Tracing ◽  
Nonlocal Nonlinear

AbstractWe study pseudo-arclength continuation methods for both Rydberg-dressed Bose-Einstein condensates (BEC), and binary Rydberg-dressed BEC which are governed by the Gross-Pitaevskii equations (GPEs). A divide-and-conquer technique is proposed for rescaling the range/ranges of nonlocal nonlinear term/terms, which gives enough information for choosing a proper stepsize. This guarantees that the solution curve we wish to trace can be precisely approximated. In addition, the ground state solution would successfully evolve from one peak to vortices when the affect of the rotating term is imposed. Moreover, parameter variables with different number of components are exploited in curve-tracing. The proposed methods have the advantage of tracing the ground state solution curve once to compute the contours for various values of the coefficients of the nonlocal nonlinear term/terms. Our numerical results are consistent with those published in the literatures.

scholarly journals Groundstates for Choquard type equations with Hardy–Littlewood–Sobolev lower critical exponent

2019 ◽  
Vol 150 (3) ◽  
pp. 1377-1400 ◽  
Author(s):  
Daniele Cassani ◽  
Jean Van Schaftingen ◽  
Jianjun Zhang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Riesz Potential ◽  
Type Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Choquard Equation ◽  
Resonant Case ◽  
Image Position ◽  
Nonlocal Nonlinear

AbstractFor the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$where $N\ges 3$, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and $I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided $N \ges 4$ or in dimension N = 3 if in addition 3/2 < α < 3 and $\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.

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