theory has three major advantages over Kalnajs’s formulation. Firstly, all eigenmodes relevant to a prescribed azimuthal wavenumber are obtained at once with classical linear algebraic algorithms. This makes it possible to explore and classify all families of growing modes beside pure oscillatory van Kampen modes. Secondly, the constituting integrals of the elements of $M_{pq}$, $C_{pq}$ and $K_{pqr}$ (Appendix \[app::Petrov-Galerkin\]) are regular at exact resonances when the condition $l\Omega_R+m\Omega_\phi-\omega=0$ holds. Finally, nonlinear interaction of modes, and the mass and angular momentum exchange between them, can be readily monitored by integrating the system of nonlinear ODEs given in (\[eq:nonlinear-ODE\]). In the proceeding section I will be concerned with the calculation and classification of modes in the linear regime.

MODES OF THE CORED EXPONENTIAL DISK {#sec:modes-exp-disk}
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JH calculated barred and spiral modes of certain stellar disks for the wavenumber $m=2$. Among the models studied in JH, the cored exponential disk with the surface density profile $$\Sigma_D(R)=\Sigma_s \exp \left ( -\lambda
\sqrt{1+R^2/R_C^2}\right ),~~\lambda=\frac{R_C}{R_D},
\label{eq:density-exp-disk}$$ and embedded in the field of the soft-centered logarithmic potential $$V_0(R)=v_0^2\ln \sqrt{1+R^2/R_C^2},
\label{eq:potential-cored-logarithmic}$$ is a viable model that resembles most features of realistic spirals. Here $R_C$ is the core radius, $R_D$ is the length scale of the exponential decay, 