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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 =6uu x +u xxx which is doubly periodic in x (i.e., the elliptic soultion) is of the form

u(x,t)=-2 i=1 N (x-x i (t))

with all x i (t) distinct except at isolated instants of time, where (z)=(z;ω 1 ,ω 2 ) is the Weierstrass functions (with periodics ω 1 , ω 2 ). The dynamics of the poles are governed by the constrained dynamical system

dx i dt=12(x i -x j ), ' (x i -x j )=0(ij)·

Any number N2 is allowed. If |ω 1 /ω 2 | is large enough and N4, 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) 2 of them coincide at t=t c and

u(x,t c )=-2 i=0 M g i (g i +1) 2(x-α i )

for some M (where N= i=1 M g i (g i -1) 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).

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35A20Analytic methods, singularities (PDE)
34M05Entire and meromorphic solutions (ODE)
37K20Relations of infinite-dimensional systems with algebraic geometry, etc.