How to use these higher symmetries and conservation laws?

Question

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation.

However, unlike the classical symmetries (point symmetries), the higher symmetries (or Lie–Bäcklund symmetries; such as KdV hierarchy) seem useless, or there are something I am unfamilar.

Similar case are the conservation laws. For KdV equation, we have infinitely many $\int u\,dx, \int u^2dx, \int \frac12u_x^2-u^3dx, \dotsc$. But it seems that only the first few conservation laws are useful.

I know some people treat the existence of infinite symmetries or conservation laws as a criterion for whether the equation is integrable, but I don't see the real application.

The question is: how to utilize these infinitely many symmetries and conservation laws?

Answer 1

One way you can use conservation laws of a PDE is in numerics; you check at each moment in time the value of the conserved quantity coming from the conservation law, to see if it is still being conserved approximately, and as soon as it is clearly not conserved (and not nearly conserved), you should no longer believe that the numerical approximation to the solution is a good approximation. The more conservation laws, the more likely that you will spot trouble in your approximation as soon as it arises.

Answer 2

Another application for higher symmetries is (hypothetical) sufficient condition for integrability.

Existence of infinitely many genuinely higher (i.e. other than point or contact and of order greater than one) symmetries is generally believed to imply that the system under study either can be transformed into a linear one (cf. e.g. the Burgers equation) or has a nontrivial Lax pair and is integrable in the sense of soliton theory.

This idea, in a somewhat modified form, when the requirement of existence of infinitely many genuinely generalized symmetries is replaced by that of existence of a nondegenerate formal symmetry (the latter is necessary for the former and significantly easier to establish) was successfully applied for classification of integrable systems by Mikhailov et al., see e.g. the recent survey Mikhailov and Sokolov - Symmetries of Differential Equations and the Problem of Integrability.

It should be stressed that the other way around this is not true: there is plenty of integrable systems, especially in the case of more than two independent variables, which possess Lax pairs and other attributes of integrability but have no genuinely higher symmetries, although they often have infinitely many nonlocal symmetries (and nonlocal conservation laws), cf. e.g. Sergyeyev - A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems, Sergyeyev - A simple construction of recursion operators for multidimensional dispersionless integrable systems, Sergyeyev - New integrable (3+1)-dimensional systems and contact geometry (published version), and Jing Ping Wang - On the structure of $(2+1)$-dimensional commutative and noncommutative integrable equations, and references therein.