Quantum Sine(h)-Gordon Model and Classical Integrable Equations

We study a family of classical solutions of modified sinh-Gordon equation, $\partial_z\partial_{\bar z} η-\re^{2η}+p(z)\,p({\bar z})\ \re^{-2η}=0$ with $p(z)=z^{2α}-s^{2α}$. We show that certain connection coefficients for solutions of the associated linear problem coincide with the $Q$-function of the quantum sine-Gordon $(α>0)$ or sinh-Gordon $(α<-1)$ models.

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