Propagation of semiclassical wave packets through avoided eigenvalue crossings in nonlinear Schrödinger equations (Q2800014)

The paper addresses a system of two nonlinear Schrödinger equations coupled by linear and nonlinear terms: NEWLINE\[NEWLINE i\epsilon\Psi_t+(\epsilon^2/2)\Psi\{xx\}-V\Psi=\kappa\epsilon^{3/2}|\Psi|^2\Psi,NEWLINE\]NEWLINE where \(\Psi\) is a two-component wave function, \(|\Psi|^2\) is the sum of squared absolute values of its two components, matrix \(V\) is built of two rows, \((x,\delta)\) and \((\delta,x)\), where \(\delta\) is a small constant, and the small parameter \(\epsilon\) implies the consideration of the semi-classical limit. In the adiabatic approximation, the gap in the system's spectrum is \(2\sqrt{\delta^2 +x^2}\). It becomes small making the spectrum close to one with a crossing of different branches at small values of \(x^2\). It is demonstrated that, when a propagating wave packet is approaching the avoided eigenvalue-crossing point, the adiabatic representation of the packet in the form of two components independently governed by the two different branches of the dispersion relation breaks down. While the nonlinear effects may be important far from the avoided-crossing point, close to it the linear terms are dominant ones, and probability of transitions between the different branches is provided by the Landau-Zener formula known in linear quantum mechanics.