# Hamiltonian of the Hirota Equation ## The cubic Nonlinear Schrodinger equation Writing the NLSE in the following form (I will use the standard $x, t$ convention here) $ i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2\psi = 0 $ We can write down the Lagrangian density: $ \mathcal{L} = i (\psi^* \psi_t - \psi \psi^*_t) - \frac{1}{2} \psi^*_x \psi_x - \frac{1}{2} |\psi|^4 $ Which allows us to write down the canonical momentum conjugate to $\psi$: $ \Pi = \frac{\partial \mathcal{L}}{\partial (\partial_t \psi)} = i \psi^* $ Now this allows us to write the well-known Hamiltonian density for the cubic NLS: $ \mathcal{H} = \frac{1}{2} \psi_x^* \psi_x - \frac{1}{2} |\psi|^4 $ ## The Hirota Equation Writing the Hirota equation in the standard way as $ i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2\psi - i \alpha \left(\frac{\partial^3\psi}{\partial x^3}+ 6 |\psi|^2 \psi \right) = 0 $ We can write down this Hamiltonian density (I got it by trail and error and consulting some literature on complex modified KdV equations): $ \mathcal{H} = \frac{1}{2} \psi_x^* \psi_x - \frac{1}{2} |\psi|^4 - i \frac{\alpha}{2} \left(\left[ \psi^*_x \psi_{xx} - \psi_x \psi^*_{xx}\right] + 3 \psi^* \psi \left[\psi \psi^*_x - \psi^* \psi_x\right]\right) = \mathcal{H}_a + \mathcal{H}_b + \mathcal{H}_c + \mathcal{H}_d $ We can check that this generates the Hirota equation via Hamilton's equations of motion: $ \psi_t = \frac{\delta \mathcal{H}}{\delta (i \psi^*)} $ It is annoying to do by hand as usual but necessary: $ \begin{align} \frac{\delta \mathcal{H_a}}{\delta \psi^*} &= -\frac{1}{2} \psi_{xx},\\ \frac{\delta \mathcal{H_b}}{\delta \psi^*} &= - \psi^* \psi^2 = -|\psi|^2 \psi, \\ \frac{\delta \mathcal{H_c}}{\delta \psi^*} &= + i \alpha \psi_{xxx} \\ \frac{\delta \mathcal{H_d}}{\delta \psi^*} &= + 6 i \alpha \psi^* \psi \psi_x = 6 i \alpha |\psi|^2 \psi_x \end{align} $ And we get the Hirota equation as expected. To the best of my knowledge, this Hamiltonian is not found anywhere in the literature. We can also check our work explicitly using Mathematica. ![[Pasted image 20210217195825.png]] ## The Nonlinear Schrodinger Hierarchy The nonlinear Schrodinger hierarchy is also a Hamiltonian system. How do we write down its Hamiltonian? The Hierarchy can be written as: ![[Pasted image 20210217200013.png]] ![[Pasted image 20210217200024.png]]