Shallow water equation
\begin{eqnarray}
\rho_t+q_x=0\\
q_t + \left(q^2 +\frac{\rho ^2}{2} \right)_x=0
\end{eqnarray}
being a strictly hyperbolic equation does not admit delta shock solutions.
Where as some strictly hyperbolic systems such as Chaplygin gas equation given by
\begin{eqnarray}
\rho_t +q_x=0\\
q_t+\left( \frac{q^2-1}{\rho} \right)_x=0
\end{eqnarray}
admits delta shock solution.
Why is this so? What is the difference between these two equations?.
Why the procedure to get solution of the Riemann problem as a combination of shock and rarefaction for strictly hyperbolic equation as explained in "Numeric approximations of hyperbolic systems of conservation laws" by Godlewsky and Raviart, does not work for this equation?
Best Answer
I have no idea if your claims are true, and I am not a specialist of the two systems above. Nevertheless, one can make the following observations.
For the first system ("shallow water"), the eigenvalues of the flux's Jacobian matrix are $\lambda_\pm = q \pm \sqrt{\rho + q^2}$. Therefore the system is strictly hyperbolic if $\rho + q^2 > 0$. The gradient $ \nabla \lambda_\pm = \pm\left(\tfrac12, \lambda_\pm \right)^\top / \sqrt{\rho + q^2} $ of the eigenvalue $\lambda_\pm$ is never orthogonal to the corresponding right eigenvector $(-\lambda_\mp/\rho, 1)^\top$ over the domain of hyperbolicity. Therefore, both characteristic fields are genuinely nonlinear, and the Riemann solution is a combination of shocks and rarefaction waves.
For the second system ("Chaplygin gas equation"), the eigenvalues are $(q\pm 1)/\rho$. Therefore, the system is strictly hyperbolic if $0 \neq |\rho| < +\infty$. The gradient $ \nabla \lambda_\pm = \left(-\lambda_\pm, 1 \right)^\top /\rho $ of the eigenvalue $\lambda_\pm$ is always orthogonal to the corresponding right eigenvector $\left(1/\lambda_\pm, 1 \right)^\top$ over the domain of hyperbolicity. Therefore, both characteristic fields are linearly degenerate. You may find the articles (1-2) interesting, which may present a similar problem.
(1) H. Cheng, "Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type", Adv. Math. Phys. (2013), 958120 doi:10.1155/2013/958120
(2) D.-X. Kong, C. Wei, "Formation and propagation of singularities in one-dimensional Chaplygin gas", Journal of Geometry and Physics 80 (2014), 58-70 doi:10.1016/j.geomphys.2014.02.009