`Positon' and `dromion' solutions of the (2+1) dimensional long wave -- short wave resonance interaction equations (Q2718273)

`Positon' and `dromion' solutions are derived for the long wave -- short wave interaction equations in a two-layer fluid. Positons are new, exact solutions of nonlinear evolution equations that exhibit algebraic decay in the far field. A positon solution can be generated by taking a special limit of multi-soliton expansion. Variation of the limiting process yields different solutions. Dromions are exact, localized solutions of \((2+1)\) dimensional (2 spatial, 1 temporal) nonlinear evolution equations that decay exponentially in all directions. One and higher dromion solutions are investigated, and a particular case of higher dromion solutions is considered in detail. By applying another limiting procedure a new solution is generated. This method of `coalescence of eigenvalues' or `wavenumbers' is thus quite universal, and can be applied to a wide variety of expansions, not just the multi-soliton type. Finally, the case of propagation solutions on a continuous wave background is also studied.