A possibility to represent nonlinear wave equations by using the ``action-angle'' variables (Q2740289)

The author analyzes the second-order equations \(d^2u/dz^2=F(u,du/dz)\) which are obtained after reduction of KdV, sine-Gordon, Kolmogorov-Petrovskij-Piskunov, and Burgers equations to the subspace of the traveling wave solutions of the form \(u(x-vt)\) where \(x\) and \(t\) are respectively the space and time independent variables, and \(v\) is the velocity of the traveling wave. All these reduced equations have homoclinic or heteroclinic trajectories each of which determines the ``action'' \(I=\int_{-\infty }^{\infty }(du/dz)^2 dz\). Using this fact the author announces the following unexpected and strange ``result'': each reduced equation can be transformed to the form \(dI/dz=0, d\Theta /dz=v\).