Exact solution of two coupled transport equations

Question

I want to solve the following system $$\eqalign{ & y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr & z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr & y(0,x) = y_0,\,\,z(0,x) = z_0, \cr} $$ I have tried to solve explcitly the fisrt and the second, but the problem is in the coupling, if the coupling is just on one of them there will be no problems. I have tried also to compute the associated semigroup to the system but I didn't get a result. Do you know any method to deal with such a system? Thank you.

Answer 1

Define the vector $X=(y,z)$ and matrices $\sigma_1={{0 1}\choose{1 0}}$, $\sigma_3={{1\; 0}\choose{0\, -1}}$, then the differential equations read $$X_t = -\sigma_3 X_x + \sigma_1 X.$$ No boundary conditions were specified at $x=0$ and $x=1$, depending on the boundary conditions you could now write the $x$-dependence as a Fourier sine or cosine series. Let me here consider an infinite range for $x$, so no boundary condition is needed. Fourier transformation $X(k,t)=\int e^{ikx} X(x,t)\,dx$ with respect to $x$ gives $$X_t = (ik\sigma_3+ \sigma_1)X\Rightarrow X(k,t)=e^{(ik\sigma_3+ \sigma_1)t}X(k,0).$$ In components this becomes $$\begin{pmatrix}y(k,t)\\ z(k,t)\end{pmatrix}=\left( \begin{array}{cc} \cosh \left(q t\right)+(ik/q)\sinh \left(q t\right)& (1/q)\sinh \left(q t\right)\\ (1/q)\sinh \left(q t\right) & \cosh \left(q t\right)-(ik/q) \sinh \left(q t\right) \\ \end{array}\right)\begin{pmatrix}y(k,0)\\ z(k,0)\end{pmatrix},$$ where I abbreviated $q=\sqrt{1-k^2}$. Notice that the matrix is even in $q$, so there is no ambiguity in the sign of the square root.