A nonlinear oscillator

Exercise (Nonlinear oscillator for a complex field). Consider a field $\psi\in\mathbb{C}$ satisfying the equation $$\begin{equation} \label{eq:oscillator} i \frac{d\psi}{dt} = -\omega\psi + g |\psi|^2\psi. \end{equation}$$ (a) Find analytically a solution of this equation when (i) $g=0$ and (ii) $g\neq 0$. [Hint. Note that $d|\psi|^2/dt=0$.]
(b) Show by explicit calculation that Eq. $\eqref{eq:oscillator}$ can be written as $$i \frac{d\psi}{dt} = \frac{\partial E}{\partial \psi^*}$$ where star (*) denotes complex conjugation and $E$ is an energy function, $$\begin{equation} \label{eq:energy} E = \frac{g}{2} |\psi|^4 - \omega|\psi|^2. \end{equation}$$ (c) Write explicitely the equation $$\begin{equation} \label{eq:dissipative} i \frac{d\psi}{dt} = (1-i) \frac{\partial E}{\partial \psi^*} \end{equation}$$ and prove that $dE/dt \leq 0$.
(d) Write code to solve numerically Eq. $\eqref{eq:dissipative}$ with arbitrary initial condition $\psi(t=0)=\psi_0$ and relate your result to the solution found in (a). [Hint. You may use Euler's method for numerical integration.]

Questions

• What would change if instead of (1-i) in Eq. $\eqref{eq:dissipative}$ we write simply i?

Learning outcomes

• Dynamics with a complex variable.
• Nonlinear oscillations.
• Basic numerical simulation.