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Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods?

Where analytical methods I refer to methods such as power series or any methods that use special and elementary functions in some form, as opposed to numerical methods which use iterations or difference schemes etc.

Thanks in advance.

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  • $\begingroup$ After two years' study I am still struggling with Cartan's equivalence method, see e.g. Olver. But if you are looking for a result more or less algorithmic, easy and immediately applicable, I think the search is pretty hopeless. $\endgroup$
    – Miguel
    Aug 23, 2015 at 16:11
  • $\begingroup$ Olver has another book, which was published in Springer called: Applications of Lie Groups to Differential Equations. Does he discuss there the same material as in the book in your link? $\endgroup$
    – Alan
    Aug 23, 2015 at 17:34
  • $\begingroup$ I am not sure because I have read neither of them... yet. I am still with the "soft" introduction by Hydon $\endgroup$
    – Miguel
    Aug 23, 2015 at 17:37

1 Answer 1

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As far as (local) power series solutions go (i.e., in the analytic category) the main existence theorem is the Cauchy-Kowalewski Theorem (in the determined, non characteristic case) and its generalization, the Cartan-Kähler Theorem (in the (possibly overdetermined) involutive case). There are further generalizations that weaken the involutivity hypothesis somewhat, such as $2$-acylcity (in the works of the Spencer school) or weighted involutivity (in the works of the Japanese school), that are useful. In all of these cases, the power series solution can be computed recursively and effectively, so in that sense, the power series solution is 'explicit'.

However, most people don't think of power series solutions as 'explicit analytic solutions'. Instead, what they usually mean is something like writing the classical solution of the wave equation $u_{xy}=0$ in the 'explicit' form $u(x,y) = f(x) + g(y)$, involving 2 arbitrary functions of one variable (each). There are nonlinear equations that admit such representations of solutions, perhaps the most famous being that of the Liouville equation $u_{xy} = e^{2u}$, which can be written in the form $$ u(x,y) = \frac12\log\left(\frac{f'(x)g'(y)}{(f(x)+g(y))^2}\right). $$

However, as Lie showed, the equation $u_{xy} = F(u)$ admits a 'general' solution in the form $$ u(x,y) = U\bigl(x,y,f(x),g(y),f'(x),g'(y),\ldots,f^{(n)}(x),g^{(n)}(y)\bigr) $$ for some given function $U$ of $2n{+}4$ variables with $f$ and $g$ being 'arbitrary' functions of a single variable, only when $F(u) = ae^{bu}$ for some constants $a$ and $b$. Note that, by Lie's theorem, even the linear equation $u_{xy} = u$ does not have such a representation. In fact, equations that admit solutions with such representations are extremely rare, and this is why modern PDE does not attempt to rely on them.

In the 1870s, Gaston Darboux developed a general method for determining when equations of various classes have representations of the above form. For example, Darboux' Method could, in theory (the computations can be formidable), determine when a given second order equation of the form $$ F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0 $$ admits an explicit solution in a form that considerably generalizes Lie's form as described above. This is probably the most advanced result of this kind about explicit analytic representations, even today, and it is quite useful in geometric situations. For example, when Darboux' Method is applied to the minimal surface equation, an elliptic nonlinear equation of the above form, it yields the Weierstrass representation for minimal surfaces in terms of a holomorphic function of a single complex variable.

One place where you can read about Darboux' Method is in a couple of papers by myself, Phillip Griffiths, and Lucas Hsu, Hyperbolic exterior differential systems and their conservation laws, I and II, Selecta Mathematica 1 (1995). For further study of Darboux' Method, there are works by Anderson, Fel, and Vassiliou, among others, that can be profitably consulted. As was suggested in the comments above, if you pursue this, you will eventually need to understand Cartan's Equivalence Method, which is the principal tool needed in the study of geometric invariants of PDE, i.e., properties of PDE that are invariant under all changes of coordinates.

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  • $\begingroup$ In general is there a way to find the explicit solution using Darboux's method? $\endgroup$ Nov 1, 2016 at 19:37
  • $\begingroup$ Darboux' Method does not always succeed. For example, for the equation $z_{xy} = z$, Lie showed that Darboux' method does not succeed in finding an explicit solution. In fact, even for linear equations of the form $z_{xy} = \lambda(x,y) z$ (where $\lambda(x,y)$ is a given function on the domain in question), it is nontrivial to determine the $\lambda$ so that the equation is solvable by the method of Darboux. $\endgroup$ Nov 1, 2016 at 20:50
  • $\begingroup$ If we know that an equation has explicit solutions (for example there is a special explicit solution by observation), how could we find the general form of all explicit solutions? $\endgroup$ Nov 2, 2016 at 1:13

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