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Zbl 0970.35130
Deconinck, Bernard; Segur, Harvey
Pole dynamics for elliptic solutions of the Korteweg-de Vries equation.
(English)
[J] Math. Phys. Anal. Geom. 3, No.1, 49-74 (2000). ISSN 1385-0172; ISSN 1572-9656/e

Any meromorphic solution of the KdV equation $u_t= 6uu_x+ u_{xxx}$ which is doubly periodic in $x$ (i.e., the elliptic soultion) is of the form $$u(x,t)= -2 \sum^N_{i=1} \wp(x- x_i(t))$$ with all $x_i(t)$ distinct except at isolated instants of time, where $\wp(z)= \wp(z; \omega_1, \omega_2)$ is the Weierstrass functions (with periodics $\omega_1$, $\omega_2$). The dynamics of the poles are governed by the constrained dynamical system $${dx_i\over dt}= 12 \sum\wp(x_i- x_j),\quad \wp'(x_i- x_j)= 0\qquad (i\ne j).$$ Any number $N\ne 2$ is allowed. If $|\omega_1/\omega_2|$ is large enough and $N\ge 4$, then nonequivalent configurations satisfying the constraint exist that do not flow into each other. The $x_i$ are allowed to coincide only in triangular numbers: if some of the $x_i$ coincide at $t= t_c$, then ${g_i(g_i+ 1)\over 2}$ of them coincide at $t= t_c$ and $$u(x, t_c)= -2\sum^M_{i=0} {g_i(g_i+ 1)\over 2} \wp(x- \alpha_i)$$ for some $M$ (where $N=\sum^M_{i= 1} {g_i(g_i- 1)\over 2}$). Explicit solutions with $N=4$ are presented with figures displaying the motion of poles $x_i(t)$ and the shape of the solution $u(x,t)$.
[Jan Chrastina (Brno)]
MSC 2000:
*35Q53 KdV-like equations
35A20 Analytic methods (PDE)
34M05 Entire and meromorphic solutions
37K20 Relations with algebraic geometry, etc.

Keywords: Korteweg-de Vries equation; pole dynamics; elliptic solution; Weierstrass functions

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