Whitham deformations of top-like integrable dynamical systems
The note is devoted to the finite-gap integration method for the equations \[ dL(\lambda)/dt+ [L(\lambda),A(\lambda)]= 0 \] in the Lax form employing the Baker-Akhiezer function \(\Psi\), a solution of the system \(L(\lambda)\Psi= \mu\Psi\), \(d\Psi/dt= A(\lambda)\Psi\). This \(\Psi\) can be found in terms of theta functions of the Riemann surface \(\Gamma\) determined by \(\text{det}(L(\lambda)= \mu I)= 0\). It is possible to introduce deformations that do not destroy neither the Liouville torus \(\text{Jac }\Gamma\) nor the quasi-periodic flow on them. These deformations are described by using the branch points \(\lambda_ k\) of \(\Gamma\) and are determined by the Whitham equation \[ (\tau d\Omega(p,\tau)+ d\Omega_ n(p,\tau))_{\lambda= \lambda_ j(\tau)}\partial_ \tau \lambda_ j(\tau)= 0, \] where \(\Omega\), \(\Omega_ n\) are appropriate Abelian integrals.
scientific article; zbMATH DE number 652156