Nonlinear Schrödinger equation with a Dirac delta potential: finite difference method*

Bin Cheng(程彬),Ya-Ming Chen(陈亚铭),Chuan-Fu Xu(徐传福),Da-Li Li(李大力),Xiao-Gang Deng(邓小刚)

Table 1. Coefficients that determine the difference scheme (18) for the cases ${N}_{1}-1\leqslant j\leqslant {N}_{1}+1$.

s = 1 s = 2 s = 3 ${d}_{s,1}(x,y)$ $\tfrac{1}{20x+4y}$ $\tfrac{4{xy}-3{y}^{2}}{3x(x+y)(3x+y)}$ $\tfrac{2{y}^{2}}{(x+y)(x+3y)(x+5y)}$ ${d}_{s,2}(x,y)$ $-\tfrac{7x+2y}{6{x}^{2}+2{xy}}$ $\tfrac{-12{xy}+3{y}^{2}}{x(x+y)(x+3y)}$ $-\tfrac{4x+5y}{4{xy}+4{y}^{2}}$ ${d}_{s,3}(x,y)$ $\tfrac{5x+4y}{4{x}^{2}+4{xy}}$ $-\tfrac{3{x}^{2}-12{xy}}{y(x+y)(3x+y)}$ $\tfrac{2x+7y}{2{xy}+6{y}^{2}}$ ${d}_{s,4}(x,y)$ $-\tfrac{2{x}^{2}}{(x+y)(3x+y)(5x+y)}$ $\tfrac{3{x}^{2}-4{xy}}{3y(x+y)(x+3y)}$ $-\tfrac{1}{4x+20y}$