Anharmonic oscillator differential equation solution. May 10, 2018

Our problem is to solve
y’’-y+2 (y^3)=0
This corresponds to a Newtonian equation of motion
1/2 (((y'))^2) + V(y) = E
V(y)= 1/2 ( -(y^2)+(y^4))

See  Eq. 25 in class notes of May 8,10

The integral of the quartic is a well known elliptic function. We do not study it in this course, but it is a very well known function at an advanced level.

Integrate[1/Sqrt[A/4+y^2-y^4],y]

To better understand these let us plot the potential V(y)= -y^2/2+y^4/2

Plot[-y^2/2+y^4/2,{y,-1.2,1.2},LabelStyle->Red,BaseStyle->{FontSize->18,FontFamily->"Helvetica"} ,Frame->True, FrameLabel->{y,"V(y)"},PlotRange->All]

The function u is the radical, i.e. y’= (SubMinus[+]) (u^(1/2))(y,E). I wrote A in place of E,because Mathematica is fussy about the symbol E, which is “reserved”.

u[y_,A_]=Sqrt[A+y^2-y^4];

For a given A, the largest value of y that keeps u real is given by g(A) which is given next.

g[A_]=Sqrt[1/2+Sqrt[A+1/4]]

In the 4 figures below we plot these s

pl[A_]:=Plot[u[y,A],{y,-g[A],g[A]},PlotRange->All]

Show[{pl[.4],pl[.2],pl[0]},LabelStyle->Red,BaseStyle->{FontSize->18,FontFamily->"Helvetica"} ,Frame->True, FrameLabel->{y,"(SuperStar[y])(E)"},PlotRange->All]

Plot[Re[u[y,-.1]],{y,-1.,1.},LabelStyle->Red,BaseStyle->{FontSize->18,FontFamily->"Helvetica"} ,Frame->True, FrameLabel->{y,"(SuperStar[y])(E)"},PlotRange->All]