# Solve an equation with real parameters

I am trying to solve $$\left(\eta ^2-1\right) \cosh ^2(\beta \eta )+2 J^2=2 \eta (J \sinh (\beta \eta ) \sinh (\beta J)+\eta \cosh (\beta \eta ) \cosh (\beta J))$$ for $$\eta$$. Where, $$\beta, J$$ are positive and reals, and $$B$$ is real. I used

Reduce[2 J^2 + (η^2 - 1) Cosh[ β η]^2 ==
2 η (η Cosh[J β] Cosh[β η] +
J Sinh[J β] Sinh[β η]) && η > 0 &&
J > 0 && β > 0, η, Reals]


• I suspect that this might be a transcendental equation with no analytic solution. Jul 3, 2020 at 6:50

ContourPlot3D shows the solution in parameterspace:
ContourPlot3D[2 J^2 + (\[Eta]^2 - 1) Cosh[\[Beta] \[Eta]]^2 ==2 \[Eta] (\[Eta] Cosh[J \[Beta]] Cosh[\[Beta] \[Eta]] +J Sinh[J \[Beta]] Sinh[\[Beta] \[Eta]]), {J, 0, 3}, {\[Beta], 0, 5}, {\[Eta], 0, 5}, AxesLabel -> Automatic ]

There are two solutions for J>1/Sqrt and one for 0<J<1/Sqrt • @yarchik The parameterspace {J, \[Beta], \[Eta]} is threedimansional, that's why I prefer ContourPlot3D . Jul 3, 2020 at 14:39
• You are right. However, I was wondering how many solution branches are there. ContourPlot[(2 J^2 + (η^2 - 1) Cosh[ β η]^2 == 2 η (η Cosh[J β] Cosh[β η] + J Sinh[J β] Sinh[β η]))/.{J->1/Sqrt}, {η,-100,100},{β,-100,100}] does not give me any. Jul 3, 2020 at 14:45
• It's necessary to evaluate the argument of ContourPlot. Try ContourPlot[ Evaluate[2 J^2 + (\[Eta]^2 - 1) Cosh[\[Beta] \[Eta]]^2 == 2 \[Eta] (\[Eta] Cosh[J \[Beta]] Cosh[\[Beta] \[Eta]] + J Sinh[J \[Beta]] Sinh[\[Beta] \[Eta]]) /. J -> 1/Sqrt] , {\[Eta], 0, 10}, {\[Beta], 0, 5}, FrameLabel -> Automatic] , this gives one branch. Jul 3, 2020 at 15:07