Modulational instability for a logarithmic nonlinear Schrödinger equation
The author considers the extended logarithmic nonlinear Schrödinger equation (NLSE) in one-dimensional case for the free particle Hamiltonian of mass \(m\): \[ i\hbar\left[\dfrac{\partial}{\partial t}+\kappa(\ln |\psi(x,t)|^2-\langle \ln |\psi(x,t)|^2\rangle)-\dfrac{\Gamma}{2}\left(\ln\dfrac{\psi(x,t)}{\overline{\psi}(x,t)}-\left\langle\ln\dfrac{\psi(x,t)}{\overline{\psi}(x,t)}\right\rangle\right) \right]\psi(x,t)=H\psi(x,t) \] Here \(\kappa\in \mathbb R\) and \(\Gamma\in \mathbb R\) are coupling strengths, the brackets \(\langle\rangle\) denote the mean with \(|\psi(x,t)|^2\) over the entire domain. Assuming the perturbations being carrier wave forms with a wave number \(\delta\) and an angular frequency \(\nu\), the author shows that the modulational instability arises when \(\nu=\dfrac{\hbar k}{m}\delta\pm \sqrt{\left(\dfrac{\hbar \delta}{2m}\right)^2-\Gamma^2}\) is a complex number. It means that the perturbation grows exponentially. If \(\delta^2\approx 0\), then the growth rate \(\Gamma\sqrt{1-\left(\dfrac{\hbar}{2m\Gamma}\right)^2\delta^2}\) approximately equals to the friction coefficient \(\Gamma\). The instability increases faster at lower frequencies of perturbation. It is indicated that the modulational instability feature of the present logarithmic NLSE differs from the cubic one, \(i\psi_t+p\psi_{xx}+q|\psi|^2\psi=0\). The presented situation is similar to the polarization instability of a light pulse propagating.
logarithmic nonlinear Schrödinger equation
scientific article; zbMATH DE number 6477334