In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are usually written in books and papers but I do not know how to find them. I tried to see how these laws were found but I couldn't. In particular, I tried the KdV equation and some of its higher nonlinear hierarchy. I have read many books about this matter but I feel I do not fully understand the practical importance and some of its related mathematics. Could you please explain to me the general picture? I will handle the technicalities. Thanks in advance.



Hamiltonian, energy, and conservation laws of nonlinear PDEs
There is a great variety of methods to obtain conservation laws of nonlinear evolution equations. In a broad classification one can divide these in symmetry-based approaches ([Noether's theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem) relates a symmetry to a conserved quantity) and direct approaches. From a practical perspective, if you wish to learn "how to find the conservation laws", the direct method using computer algebra tools seems the most efficient.


The direct method is more general than a method based on Noether's theorem, because it does not require that the linearization of the evolution equation be self-adjoint. If you wish to restrict yourself to self-adjoint linearizations the procedure is to identify the action functional which is minimized by the evolution. Each point symmetry of the action is then associated with a conservation law, and Noether's theorem gives an explicit formula to obtain it.


These two approaches are well explained in [Construction of conservation laws: how the direct method generalizes Noether’s theorem](https://personal.math.ubc.ca/~bluman/cyprus%20proceedings%20paper.pdf). The software to implement the direct method is described in [Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations.](https://arxiv.org/abs/0803.0083)


