Unstable modes of non-axisymmetric gaseous discs

We present a perturbation theory for studying the instabilities of non-axisymmetric gaseous discs. We perturb the dynamical equations of self-gravitating fluids in the vicinity of a non-axisymmetric equilibrium, and expand the perturbed physical quantities in terms of a complete basis set and a small non-axisymmetry parameter $\epsilon$. We then derive a linear eigenvalue problem in matrix form, and determine the pattern speed, growth rate and mode shapes of the first three unstable modes. In non-axisymmetric discs the amplitude and phase angle of travelling waves are functions of both the radius $R$ and the azimuthal angle $\phi$. This is due to the interaction of different wave components in the response spectrum. We demonstrate that wave interaction in unstable discs with small initial asymmetries, can develop dense clumps during the phase of exponential growth. Local clumps, which occur on the major spiral arms, can constitute seeds of gas giant planets in accretion discs.