N-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 bilinear form, \[ {\begin{array}{llll} (-4D_xD_t+D_x^4+3D_y^2) τ_n\cdotτ_n=24τ_{n-1}τ_{n+1}, (2D_t+D_x^3\mp 3D_xD_y) τ_{n\pm 1}\cdotτ_n=0 \end{array} \quad n=1,2,.... \] where ${\tau }_0=1$. The $\tau$-functions ${\tau }_n$ are given by the pfaffians of certain skew-symmetric matrix. We identify one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by $4N\times 4N$ skew-symmetric constant matrix which we call the $B$-matrices. We then find that one can have $M$-soliton solutions with $M$ being any number from $N$ to $2N-1$ for some of the $4N\times 4N$ $B$-matrices having only $2N$ nonzero entries in the upper triangular part (the number of solitons obtained from those $B$-matrices was previously expected to be just $N$).

22 pages, 12 figures

Cited by in corpus (6)

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• Geometry of the Pfaff lattices
• Mach-Type Soliton in the Novikov-Veselov Equation
• Multi-variable reductions of the dispersionless DKP hierarchy
• Loewner equations and reductions of dispersionless hierarchies