Characteristics of equations of the form $u_{xy}=f(u_x,u_y,u)$

Question

In the usual treatment of hyperbolic differential equations, it is always assumed that there are two families of characteristics. That is, if the equation $L[u]-f(u_x,u_y,u)=au_{xx}+2bu_{xy}+cu_{yy}-f(u_x,u_y,u)=0$ is hyperbolic, by definition the roots $\zeta_{\pm}$ of the polynomial $q(\zeta)=a\zeta^2-2b\zeta+c$ are real, and then one defines the characteristic curves by means of the ODEs $dy/dx=\zeta_{\pm}(x,y)$. But for example for the equation $u_{xy}=u$, the only root is $\zeta_+=\zeta_-=0$, then we would have only one family of characteristics so the equation is not hyperbolic but parabolic, but this contradicts the fact that a equation is parabolic iff $b^2-ac=0$ and hyperbolic iff $b^2-ac>0$.

What's the true nature of the equations of the form $u_{xy}=f(u_x,u_y,u)$, and what are the implications of the fact that only one set of characteristics exists, even though it's supposed to be a hyperbolic equation?

Answer 1

This question has already been answered by Han de Bruijn: however I feel I can make explicit some conceptual points and give a few references and insights.

• First of all let me point out the most important thing: Han has shown by an example that the following equation where $L$ is a linear hyperbolic operator $$ L[u]-f(u_x,u_y,u)=au_{xx}+2bu_{xy}+cu_{yy}-f(u_x,u_y,u)=0 \label{a}\tag{HE} $$ has two (almost, see the notes) equivalent normal forms, i.e. $$ \begin{align} u_{\eta\xi} & = f_1(u_\eta,u_\xi,u) \label{1}\tag{1}\\ u_{\eta\eta}-u_{\xi\xi} & = f_2(u_\eta,u_\xi,u) \label{2}\tag{2} \end{align} $$ To see this, let's recall the basic effect of coordinate transformation applied to the linear part of \eqref{a}: given a one-to-one map $$ \begin{cases} \eta=\eta(x,y)\\ \xi = \xi(x,y) \end{cases} \implies J \begin{pmatrix} \eta & \xi \\ x & y \end{pmatrix}\neq 0 $$ we have that $$ \begin{split} L(u)=au_{xx}+2bu_{xy}+cu_{yy} &= [a\eta_x^2+2b\eta_x\eta_y+c\eta_y^2] u_{\eta\eta}\\ &\; + 2[a\eta_x\xi_x+b(\eta_x\xi_y+\eta_y\xi_x)+c\eta_y\xi_y] u_{\eta\xi}\\ &\; + [a\xi_x^2+2b\xi_x\xi_y+c\xi_y^2] u_{\xi\xi}\\ &\; + \text{ lower order linear terms in $\eta_x, \eta_y, \xi_x, \xi_y$} \end{split}\label{3}\tag{3} $$ Now consider the characteristic equation for $L$, i.e. $$ a\varphi_x^2+2b\varphi_x\varphi_y+c\varphi_y^2 = 0.\label{b}\tag{CE} $$ Providing proper condition are stated (see again the notes below), since $L$ is hyperbolic (i.e. $b^2- ac>0$) this equation has two distinct real solutions , say $\varphi_1$ and $\varphi_2$: it is possible to demonstrate that these functions define a one-to-one map. If in \eqref{3} we put $$ \begin{align} \eta (x,y) = \varphi_1 (x,y)\\ \xi (x,y) = \varphi_2 (x,y) \end{align}\label{4}\tag{choice 1} $$ we get the normal form \eqref{1}, while if we put $$ \begin{align} \eta (x,y) = \varphi_1 (x,y)+\varphi_2 (x,y)\\ \xi (x,y) = \varphi_1 (x,y)-\varphi_2 (x,y) \end{align}\label{5}\tag{choice 2} $$ we get the \eqref{2} normal form.
• The two different change of variables generate two different lower order linear terms, are that are added to the transformed nonlinear right hand side of \eqref{1} i.e. $f(u_x, u_y, u)$ to give rise to the right hand sides $f_i(u_\eta,u_\xi, u)$, $i=1,2$ of respectively \eqref{1} and \eqref{2}.
• Note that \ref{5} corresponds to the $\alpha={\pi/ 4}$ choice in the example of Han. On the other hands, it is easy to see the reason for the apparent inconsistency of \eqref{2} respect to the definition of "hyperbolic PDO": the coodinate axes $\eta=0$ and $\xi=0$ generated by \ref{4} are images of the characteristic curves.
• Why the normal form \eqref{1} was classically more used than normal form \eqref{2}?: this is simply because the solutions of \eqref{1} need to be of class $C^2$, while the solution of \eqref{2} are "only" required to be of class $C^1$ with continuous mixed derivatives, i.e. $u_{xy}\in C^0$. This may seem silly, as today the concept of solution has evolved in order to include non-differentiable-at-all functions but in the classical context formulation \eqref{2}, requiring a somewhat lower regularity, may have seen as appealing in order to ease the study of the PDE.
• For this answer I heavily used reference [2] (see chapter II, §1-2 pp. 130-140) since it deals comprehensively with this equation and many more in a classical setting. Of course the problem is dealt with in other wonderful references in English, like in [1] (see chapter I, §2.2, pp. 22-26) or [3] (see chapter III, §1.1, p. 155), however in a very concise fashion.
• The analysis of the homogeneous ultrahyperbolic equation, i.e. $$ \frac{\partial^2}{\partial x\partial y}u=u_{xy}=0 $$ is describer in this MathOverflow answer. The references given therein bring to the original works of Gaetano Fichera and Mauro Picone who studied this equation thoroughly.

References

[1] Andreĭ Vasil’evich Bitsadze, Equations of Mathematical Physics (English), Translated from the Russian by V. M. Volosov and I. G. Volosova, Moskva: Mir, p. 318 (1980), ISBN: 5-03-000533-1, MR1024787, Zbl 0499.35002.

[2] Silvio Cinquini, Maria Cinquini Cibrario, Equazioni a derivate parziali di tipo iperbolico. (Italian) Monografie Matematiche del Consiglio Nazionale delle Ricerche 12. Roma: Edizioni Cremonese, pp. VIII+552 (1964), MR0203199, Zbl 0145.35404.

[3] Richard Courant, David Hilbert, Methods of mathematical physics. Volume II: Partial differential equations. Translated and revised from the German Original. Reprint of the 1st Engl. ed. 1962. (English) Wiley Classics Edition. New York-London-Brisbane: John Wiley & Sons/Interscience Publishers, pp. xxii+830 (1989), ISBN: 0-471-50439-4, MR1013360, Zbl 0729.35001.