# ultrahyperbolic PDE

Just wondering:

1. How to solve ultrahyperbolic PDEs? Is there any analytical solution for linear ultrahyperbolic PDEs?
2. If there are only numerical solutions, are the solutions' behavior similar to those of nonlinear eqns? I mean, like an anharmonic oscillator, chaotic but deterministic?

Information of reference books is also welcomed - but better not too specialized in math. I am an engineering student with very very limited math skills.

Thanks :)

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I provided an answer below in the context of the initial value problem (since you mentioned the word "deterministic" [which by the way, is not true of the ultrahyperbolic equations]). If you mean something else, please edit the question to clarify. Note also that the ultrahyperbolic equation is closely related to the Radon and X-ray transforms. So there are also lots of highly theoretical literature about it, usually under the guise of symmetric spaces and harmonic analysis on them. – Willie Wong Dec 17 '10 at 12:49

The answer depends on what you mean by "solving" the PDE. The initial value problem for the ultrahyperbolic PDEs are ill-posed. In particular, there is a theorem (the version I know is due to Hormander, you can find it in his Analysis of Linear Partial Differential Operators; but presumably some versions go back earlier) which states that:

Theorem Let $L$ be a linear partial differential operator with smooth coefficients of order $m$ on $\mathbb{R}^{1+n}$. Consider the Cauchy problem for $Lu = F$ on the upper-half-space $\{ x_0 \geq 0 \}$ with initial data $u_0, u_1, \ldots, u_{m-1}$ (such that $(\partial_0)^k u|_{x_0 = 0} = u_k$ ). Then the following are equivalent. (a) The Cauchy problem has a unique smooth solution $u$ for every prescribed smooth data $u_0,\ldots,u_{m-1}$ and source $F$ and (b) $L$ is a hyperbolic operator.

So in particular, in general there cannot be well-defined solutions to the initial value problem for ultrahyperbolic PDEs. (They either don't exist or aren't unique. And in the case you do have a solution, the solution is unstable.)

Now, you cannot even construct approximate solutions to the initial value problem reliably using numerical methods, since ultrahyperbolic equations do not have finite-speed of propagation, so you cannot "localise" the problem, and small changes around a point $x$ may almost instantaneously affect the solution at a far-away point $y$.

In certain special cases you can produce some semblance of a solution. In the constant coefficient case in second order, where the equation can be written as $(\triangle_X - \triangle_Y)u = 0$, you have what is known as Asgeirsson's Mean Value Theorem which is sort of a generalisation of the mean value theorem for harmonic functions, and also a generalisation for the Green's function formula for the linear, constant coefficient wave equation. In this particular case you can also consider solutions using Fourier analytic methods. From there one sees that if one were to assume certain restrictions on the allowed wave-numbers (which leads to a non-local constraint on the initial data), one can recover well-posedness of the initial value problem.

As to references, perhaps an easy way to do a reverse search on the classical paper of Fritz John on the subject. John's various textbooks also contain some information about it; in particular you may want to consult his Partial Differential Equations book, and his book on Plane Waves and Spherical Means Applied to Partial Differential Equations.

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Willie, concerning your third paragraph, do you know where may I read about the Fundamental solutions to ultrahyperbolic pde?, $$\Delta_x u =\Delta_y u$$ thanks. – MathematicalPhysicist Oct 24 '12 at 11:50
Willie, I was led to believe by a seminar that I attended that Fritz John gave certain submanifolds where you can demand arbitrary values to the solution, and a complete solution to the whole space can be extended from those boundary conditions. Is that accurate, and if it is, could you extend or adjust your answer regarding that? – lurscher Jan 4 '14 at 23:49
@lurscher: well-posedness requires existence and uniqueness. Almost certainly you are thinking of the case where data is prescribed on a characteristic submanifold, which admits nonunique extensions. But of course, given your brief description I can't say anything conclusive. Perhaps you should look at John's paper on the subject. (I'm inclined to say it is also in his Plane Waves and Spherical Means... book, but don't have my copy handy to check.) – Willie Wong Jan 6 '14 at 8:59
@MathematicalPhysicist: sorry for the delayed response, somehow I have not seen your comment earlier. If you mean the fundamental solution to the initial value problem, there isn't one as the equation is not well-posed. The closest you have is Asgeirsson's Mean Value Theorem. See the link given in my previous comment (and possibly the Plane Waves... book I mentioned) for more details. – Willie Wong Jan 6 '14 at 9:03