The equations, in the absence of gravity, admits an explicit solution, which is called as the N-pole solution since the equation admits a pole decomposition,as shown by Olivier Thual, Uriel Frisch and Michel Hénon in 1988. Consider the 1d equation
where  is the Fourier transform of . This has a solution of the form
where  (which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity , the it is sufficient to consider poles whose real parts lie between the interval  and . In this case, we have
These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.