Twodimensional solitons of the Kadomtsev–Petviashvili equation and their interaction, Phys

Explicit analytic formulae for two-dimensional solitons are given. It is proved that, unlike one-dimensional solitons, two-dimensional ones do not interact at all. Non-linear quasi-one dimensional waves (withy where F is an arbitrary solution of the system of equa-much larger than x) in a weakly dispersive medium tions are described by the Kadomtsev-Petviashivili equation ~ a2F = 0, + 4 ~ a3F[1]: a—1- — — 4 ay ax2 ax’2 at ax3 ax a(u~+ 6uu~+ u~~~)/ax = — 3a2 a2u/ay2. (1) implies that function The sign of the parameter —a2 coincides with that of u(x,y,t)—2 aK(x,x,y,t)/ax (4) the dispersion parameter a2w/ak2. It was shown that eq.(1) could also be formulated obeys eq. (1). In particular, putting a = 1 and choosing for the inverse scattering problem [2, 3]. for F the form It was already noticed [1] that plane solitons are N unstable under transverse perturbations in a positive F = ~c~~’y,t) exp(pnx + q 0x’) dispersion medium (exact solution of the stability n=1 problem is given in ref. [4]). This observation led to the speculation that there are stable two-dimensional where solitons localized in the x-y-plane [51and their profiles c~(y,t) = cn(O) exp[(—p~+ q~)y — 4(p~+ q~)t] are numerically found [5] In this paper we present analytic expressions for we arrive at the degenerate kernel of eq. (2). With eq. two-dimensional soliton solutions in terms of rational (4) we obtain functions of x andy, as well as for arbitrary systems a 2 of solitons. In our study of this problem we are largely u = 2 — ln det A, (5) encouraged by ref. [61, where singular rational solu- &x2 tions of the KdV equation are found. where A is an N X N matrix We construct our exact solutions as follows [2] The equation Anm = ~nm + c 0(y, t) ~(Pn+~n)x1p~+ q ~. (6) K(x, x’,y, t) + F(x, x’,y, t) This solution has first been found in ref. [2].. It des-cribes the N-plane interaction and does not decrease in directions in the x-y-plane. x + f K(x,x”, y, t) F(x”, x’, ~, t) ~ ‘ = 0, (2) Formulae (5) and (6) holds for arbitrary complex