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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. (English) Zbl 0970.35130

Any meromorphic solution of the KdV equation ${u}_{t}=6u{u}_{x}+{u}_{xxx}$ which is doubly periodic in $x$ (i.e., the elliptic soultion) is of the form

$u\left(x,t\right)=-2\sum _{i=1}^{N}\wp \left(x-{x}_{i}\left(t\right)\right)$

with all ${x}_{i}\left(t\right)$ distinct except at isolated instants of time, where $\wp \left(z\right)=\wp \left(z;{\omega }_{1},{\omega }_{2}\right)$ is the Weierstrass functions (with periodics ${\omega }_{1}$, ${\omega }_{2}$). The dynamics of the poles are governed by the constrained dynamical system

$\frac{d{x}_{i}}{dt}=12\sum \wp \left({x}_{i}-{x}_{j}\right),\phantom{\rule{1.em}{0ex}}{\wp }^{\text{'}}\left({x}_{i}-{x}_{j}\right)=0\phantom{\rule{2.em}{0ex}}\left(i\ne j\right)·$

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 $\frac{{g}_{i}\left({g}_{i}+1\right)}{2}$ of them coincide at $t={t}_{c}$ and

$u\left(x,{t}_{c}\right)=-2\sum _{i=0}^{M}\frac{{g}_{i}\left({g}_{i}+1\right)}{2}\wp \left(x-{\alpha }_{i}\right)$

for some $M$ (where $N={\sum }_{i=1}^{M}\frac{{g}_{i}\left({g}_{i}-1\right)}{2}$). Explicit solutions with $N=4$ are presented with figures displaying the motion of poles ${x}_{i}\left(t\right)$ and the shape of the solution $u\left(x,t\right)$.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35A20 Analytic methods, singularities (PDE) 34M05 Entire and meromorphic solutions (ODE) 37K20 Relations of infinite-dimensional systems with algebraic geometry, etc.