nonlinear differential equation

Question

For such a equation

$$-y''(x) + x^2\cdot y(x) - |y(x)|^2\cdot y(x) = 0$$

with initial conditions: $y(-500) = 0$ and $y(500) = 0$.

I write in mathematica

ic1 = 0;
xmax = 500;
xmin = -500;
sol = NDSolve[{-y''[x] + x^2*y[x] - Abs[y[x]]^2*y[x] == 0, 
   y[xmax] == ic1, y[xmin] == ic1}, y[x], {x, xmin, xmax}]
Plot[y[x] /. sol, {x, xmin, xmax}]

but the graph is incorrect. Please help me

Answer 1

It seems that the ode under discussion has only trivial solution.

I checked it with both Mathematica and maple and got what was expected.

Mathematica output

Plot[y[x] /. sol, {x, xmin, xmax}]

Maple output

restart:with(plots):

ode:=-diff(y(x),x$2)+x^2*y(x)-abs(y(x))^2*y(x)=0;

ics:=y(-500)=0,y(500)=0;

p:=dsolve({ode,ics},numeric);

odeplot(p);