Time reversal symmetry

Time reversal symmetry refers to any symmetry in which every solution to a dispersive equation comes with a counterpart which evolves backwards in time compared to the original solution, thus the original solution at time t is linked to the reversed solution at time -t. While most dispersive model equations enjoy a time reversal symmetry, this symmetry manifests itself differently from equation to equation:

• For wave equations, the time-reversal of a solution $u(t,x)$ is usually $u(-t,x)$, although for tensor-valued fields one may also have to negate certain "time components" of the field.

• For Schrodinger equations, the time-reversal of a solution $u(t,x)$ is usually ${\overline {u}}(-t,x)$.

• For KdV-type equations and the Benjamin-Ono equation, the time-reversal of a solution $u(t,x)$ is usually $u(-t,-x)$.

Time reversal symmetry shows that the forward-in-time behavior of solutions is typically very similar to the backward-in-time behavior. This in turn implies that phenomena with a preferential direction of time (such as dissipation) cannot occur; on the other hand, phenomena such as local smoothing can still occur because they are a tradeoff between two characteristics of a solution (in this case, decay is traded for regularity). Note that the general phenomenon of KdV-type equations that solitons move to the right, while radiation moves to the left, does not contradict this because the time reversal symmetry in this case also reverses left and right.

Time reversal symmetry is clearly consistent with the presence of conservation laws. It is less obvious that this symmetry can be consistent with the presence of monotonicity formulae, but they are compatible if the monotone quantity changes sign whenever time is reversed. Because of this, monotone quantities are typically built out of signed quantities such as the momentum rather than from unsigned quantities such as mass or energy.