<i>N</i>-soliton solutions to the DKP equation and Weyl group actions

Abstract

We study soliton solutions to the DKP equation which is defined by the Hirota\nbilinear form, \\[ {\\begin{array}{llll} (-4D_xD_t+D_x^4+3D_y^2)\n\\tau_n\\cdot\\tau_n=24\\tau_{n-1}\\tau_{n+1}, (2D_t+D_x^3\\mp 3D_xD_y) \\tau_{n\\pm\n1}\\cdot\\tau_n=0 \\end{array} \\quad n=1,2,.... \\] where $\\tau_0=1$. The\n$\\tau$-functions $\\tau_n$ are given by the pfaffians of certain skew-symmetric\nmatrix. We identify one-soliton solution as an element of the Weyl group of\nD-type, and discuss a general structure of the interaction patterns among the\nsolitons. Soliton solutions are characterized by $4N\\times 4N$ skew-symmetric\nconstant matrix which we call the $B$-matrices. We then find that one can have\n$M$-soliton solutions with $M$ being any number from $N$ to $2N-1$ for some of\nthe $4N\\times 4N$ $B$-matrices having only $2N$ nonzero entries in the upper\ntriangular part (the number of solitons obtained from those $B$-matrices was\npreviously expected to be just $N$).\n