In[]:=
sol=Solve[q==-z^2/(2L^2),z]
Out[]=
z-
2
L
q
,z
2
L
q

2L==z(1+Tanh[z])/.sol(*Twosolutions!*)
Out[]=
2L-
2
L
q
1-Tan
2
L
q
,2L
2
L
q
1+Tan
2
L
q

In[]:=
ContourPlot2L-
2
L
q
1-Tan
2
L
q
,{L,-1/3,3},{q,-5,0},FrameFalse,AxesTrue,AxesLabelAutomatic,PlotPoints->100
Out[]=
In[]:=
ContourPlot2L
2
L
q
1+Tan
2
L
q
,{L,-1,1},{q,-1,1},FrameFalse,AxesTrue,AxesLabelAutomatic,PlotPoints->20
Out[]=