The planet-disk interactions have been studied extensively in the linear regime. What if we extend the analysis to the next order? The linear modes begin to couple and causes higher-order modes, called ultraharmonics. It is relatively simple in the case of spiral galaxies as there is only one linear mode (the bisymmetric m=2 mode). For a planet, which is a point-mass, there are many linear modes. Therefore, the mode-coupling is a little bit more complicated. My recent paper (Lee 2016) was to explore the mode-coupling among many linear modes and see if that can explain the existence of the secondary spirals found in simulation with a massive planet. It turns out that even to the second order, many spirals (m=2, 3 etc) can also be excited. This is contrary to the previous guess that two-arm spirals are caused by self coupling of linear one-arm spiral (see the slide). In any case, it is still an open question whether mode-coupling is the only nonlinear effect causing the secondary spirals.
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