# 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. – Igor Khavkine Feb 18 at 12:17
• Any ODE system trivially corresponds to a linear hyperbolic PDE for which the ODE solution curves are characteristic. – Michael Renardy Feb 18 at 13:34

## 1 Answer

When Nonlinear Differential Equations are Equivalent to Linear Differential Equations

A necessary and sufficient condition is established for the existence of a 1-1 transformation of a system of nonlinear differential equations to a system of linear equations. The obtained theorems enable one to construct such transformations from the invariance groups of differential equations. The hodograph transformation, the Legendre transformation and Lie’s transformation of the Monge-Ampère equation are shown to be special cases. Noninvertible transformations are also considered. Examples include Burgers’ equation, a nonlinear diffusion equation and the Liouville equation.

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 – Weather Report Feb 19 at 10:03