Peculiarities of resonant interactions of lump chains within the KP1 equation

Using the Hirota bilinear method, we derive resonant solutions to the KP1 equation. Solutions describe lump chains differently oriented in (x, y)-plane. We show that resonant solutions arise as the limiting case of more general non-resonant solutions when phase shifts of lump chains caused by their interaction become infinite. Resonant solutions can describe both stationary patterns (for example, Y-shaped patterns consisting of three different lump chains) and non-stationary interacting parallel lump chains. In the latter case, a lump chain can be emitted/absorbed by another lump chain. As the number of lump chains increases, resonance phenomena become more complex and diversified including the cases of exchange of a lump chain by two other lump chains. The method used in this paper can be extended to apply to other integrable systems in two and three spatial dimensions such as, for example, described by Mel'nikov's equations