Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

Question

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form

$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+\mathbf{A}\partial_{t}\mathbf{u}+\mathbf{B}\left(t\right)\mathbf{u}=0$$

with $\left(x,y,t\right)$-independent matrix $\mathbf{A}$ and $t$-dependent / $\left(x,y\right)$-independent $\mathbf{B}$. Also, $t\in\left[0,2\pi\right]$ is not really a time, but an angle, and $\mathbf{B}\left(t+2\pi\right)=\mathbf{B}\left(t\right)$ is periodic, so I guess the solutions will be periodic as well. Actually $\mathbf{A}$ is real and diagonal, so my system of equations seems to be in a canonical form from the beginning. Note nevertheless that two eigenvalues of $$\mathbf{A}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$ are degenerate, so I wonder if the system is still hyperbolic (I know it is no more strictly hyperbolic according to Courant (see references below), but Wikipedia defines it as hyperbolic / I've been unable to find the Wikipedia definition in any textbook I've put an hand on). I would like (if possible) to obtain analytic (in a close form or as a series expansion) solutions of this problem, or understand a bit better which perturbation scheme I could use. It seems to me this system is extraordinary simple, and at the same time I'm paralysed by my ignorance in powerful enough methods.

So my first and general question would be : which literature would you recommend to me ? about this problem.

Giving more details: for the moment I've found

R. Courant and P. Lax On nonlinear partial differential equations with two independent variables Commun. Pure Appl. Math. 2, 255 (1949). (beyond a paywall)

where section 3 seems of interest for me, it shortly discusses an old method by

O. Perron Über Existenz und Nichtexistenz von Integralen partieller Differentialgleichungssysteme im reellen Gebiet Math. Zeitschrift 27, 549 (1928). (also beyond a paywall)

which seems to answer my problem, giving analytical solution of system of semi-linear partial differential equation. The book by

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II: Partial Differential Equation (John Wiley and Sons, 1962).

seems too much related to non-linear systems, and I've been unable to figure out what to do with my problem. Finally a short lecture note by

E. Kersalé, Analytic Solutions of Partial Differential Equations http://www1.maths.leeds.ac.uk/~kersale/teaching.html (2003)

ends up with a short discussion of my problem, but I still do not figure out what to do with the system of ordinary differential equations once fond the characteristics lines / surfaces... (actually, two characteristics are circles, and two straight lines are degenerate). So are there other (perhaps better or more specific for physicists) textbook/notes you would recommend ?

Thanks in advance for any remark aiming at improving this question.

Answer 1

For simplicity, consider solutions where $ u$ does not depend on $x, y$: $A u_t + B(t) u = 0$. If $y^T A = 0$, that says $y^T B(t) u = 0$, so $u$ is restricted to belong to a certain (possibly $t$-dependent) subspace.
Thus for your example $$A = \pmatrix{1 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & -1\cr}$$ $(B(t) u)_2 = (B(t) u)_3 = 0$. If the appropriate $2 \times 2$ submatrix of $B(t)$ is invertible, this lets you express $u_2$ and $u_3$ in terms of $u_1$ and $u_4$, and you get a periodic linear system for $u_1$ and $u_4$.
The solutions are usually not periodic in $t$. Rather, the linear operator $u(0) \to u(2\pi)$ will have eigenvalues $\lambda$ corresponding to solutions where $u(2\pi) = \lambda u(0)$ (see Floquet theory).

Somewhat more generally, solutions of the form $u(x,y,t) = \exp(\alpha x + \beta y) v(t)$ lead to the same type of system (with $\alpha \cos(t) + \beta \sin(t)$ added to $B$).

EDIT: If the submatrix of $B(t)$ is not invertible for some $t$, you may find that some or all of the nontrivial solutions have singularities at those $t$.

Answer 2

Well, assuming that the matrix $\mathbf{B}$ is a real-analytic function of $t$, the local real-analytic theory gives you this result, which may or may not be useful to you:

Start with a real-analytic, $2$-dimensional surface $S\subset\mathbb{R}^3$ (coordinates $x,y,t$) that is nowhere tangent to any one of the three vector fields $$ E_c = \cos(t)\ \frac{\partial\ }{\partial x}+\sin(t)\ \frac{\partial\ }{\partial x} + c\ \frac{\partial\ }{\partial t} $$ where $c = 0,1,-1$ and choose a real-analytic function $\phi:S\to\mathbb{R}^4$. Then there will exist an open neighborhood $U\subset\mathbb{R}^3$ of $S$ on which there will exist a unique real-analytic function $\mathbf{u}:U\to\mathbb{R}^4$ that satisfies your equation and has the property that $\mathbf{u}(p) = \phi(p)$ for all $p\in S$.

The point is that the data $(S,\phi)$ are noncharacteristic initial data for the Cauchy problem, and, because of real-analyticity, the Cauchy-Kowalewski theorem applies.