Solve an equation with real parameters

Question

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]

with no answers.

Answer 1

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[2] and one for 0<J<1/Sqrt[2]