# Nonlinear ODE to linear PDE?

I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion.

Consider a classical mechanical system with one degree on freedom $$q$$ and a Hamiltonian $$H(q,p,t)$$. The equations of motion can be cast as the Euler-Lagrange equations which are second-order ODE, non-linear in the generic case. Yet another equivalent way is through the Hamilton-Jacobi equation $$\frac{\partial S}{\partial t}+H\left(q,\frac{\partial S}{\partial q},t\right)=0 \qquad (1)$$ which is a single non-linear PDE.

Quantum-mechanically this system is described by a linear Schrodinger equation (I gloss over all ambiguities and subtleties that might be in quantizing a classical system) $$-i\hbar\frac{\partial \psi}{\partial t}+H(q,-i\hbar\partial_q,t)\psi=0 \qquad (2)$$

Formally, equation (1) is the $$\hbar\to0$$ limit of equation (2) if one makes the following ansatz $$\psi=e^{iS/\hbar}$$. So it seems that at least formally for a large class of ODE one can find an equivalent linear PDE. I am interested to learn if there is any systematic theory to back up this heuristic arguments.

• I think you are playing a bit loose with the term "equivalent". As you point out yourself, the $\hbar\to 0$ limit is necessary to relate solutions of the Schroedinger and Hamilton-Jacobi equations. Moreover, the HJ equation is an ODE only in the stationary case with one spatial dimension. I think your question could do with some clarification. Feb 18, 2021 at 12:17
• Any ODE system trivially corresponds to a linear hyperbolic PDE for which the ODE solution curves are characteristic. Feb 18, 2021 at 13:34

I don't think the Schrödinger equation is a useful example in this class, because the correspondence between the classical and quantum dynamics breaks down after a time $$T$$ that grows only logarithmically when $$\hbar$$ is sent to zero.$$^\ast$$ So you will not be able to follow the classical dynamics for a meaningful time even if $$\hbar$$ is very small. One way in which this difficulty appears, is that the classical dynamics can be chaotic, whereas quantum dynamics is quasiperiodic.
$$^\ast$$ The time $$T$$ at which the quantum-classical correspondence breaks down is called the Ehrenfest time, it is of the order $$T=\alpha^{-1}\log\hbar$$, with $$\alpha$$ the Lyapunov exponent of the classical dynamics.
• Thanks for both links, very instructive. However I had something a bit different in mind, perhaps not articulated clearly. As far as I can tell the paper that you linked mostly considers 1-1 mappings of between linear and non-linear diff equations (with some reservations for not 1-1 mappings.) The mappings exists only for very particular types of equations. In contrast I was asking if one can trade non-linearity for linearity in a possibly much larger space. In my example $q(t)$ is a one-parameter function with a non-linear equation while $\psi(q,t)$ is now two-parameter but obeying the linear Feb 19, 2021 at 10:03