Solitary wavepackets from oscillatory non-linear equations with damping (Q437156)

In determining a family of exactly integrable differential equations showing the physical characteristics of damped non-linear differential equations of the type \[ \frac{d^2\theta}{dt^2}+f(t)\frac{d\theta}{dt}+\sin\theta=0 \] with \(f(t)\) being an arbitrary function, the author analyzes a subclass of the family of differential equations \[ \begin{aligned} \frac{d^2\theta}{dt^2}+f_0(t)\frac{d\theta}{dt}+f_1(t)\theta+2f_2^2(t)\theta^3+f_3(t)=0, \end{aligned} \] which are integrable in the sense of Painlevé. The author maps the analyzed equations into a canonical form and derives Hamiltonian and Lagrangian functions for the canonical form and the initial differential equations. The canonical form admits a typical property of harmonic damping systems: the existence of an invariant that splits into the product of other unrelated conserved quantities. For a subclass of the family, solutions fastly decay to zero, present oscillatory behavior given by elliptic functions and have a finite number of nodes. The studied family of differential equations admits applications: ac-Josephon effect, thermometry for low temperature physics, semiconductor lattices, etc. (see for instance [\textit{K. N. Alekseev} and \textit{F. V. Kusmartsev}, Phys. Lett., A 305, No. 5, 281--288 (2002; Zbl 1001.82114)]).