Soliton-like behaviour in multi-component Nonlinear Schrödinger equations
Published:
Nonlinear Schrodinger (NLS) Equation is ubiquitous in physics, still not much is known about the dynamics of the equation. Through our investigation, we found that multicomponent NLS system can be mapped to an uncoupled Korteweg-de Vries (KdV) system. Since KdV is an integrable system we can know about certain qualitative aspects of multicomponent NLS from this mapping. One interesting feature of KdV equation is that it has a special solution called solitons, these special profiles balance the nonlinear and the dispersive effects in the system to give traveling wave packets. Leveraging our mapping one can find similar solutions for multicomponent NLS systems as well.
The multicomponent NLS equation is given by
\[i \hbar \frac{\partial \psi_k}{\partial t} = \frac{\hbar^2}{2}\frac{\partial^2 \psi}{\partial x^2} - \sum_{j=1}^{j=N}\alpha_{kj}|\psi_j|^2\psi_k + G_{kk}|\psi_k|^4 \psi_k\]where $\psi_k$ is the wave macroscopic wavefunction, $\alpha$ is the matrix of coupling constants and $G_{kk}$ is the quintic self-coupling coefficient.
The KdV equation is given by
\[\frac{\partial u}{\partial t} + 6u\frac{\partial u}{\partial x} + \frac{\partial ^3 u}{\partial x^3} = 0\]This mapping is done using Method of Multiple Scales and you can read more about it in the project report. We have also developed a package which maps any input multicomponent NLS system to uncoupled KdV system and checks the robustness of the generated profile. The source code will soon be available as an open source package.
We found that these traveling wave packets are extremely robust and remain intact even after colliding with one another as shown in the figure below.
Thus we can generate robust initial conditions for multicomponent NLS system using this mapping.