zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $$u(x,t)= -2 \sum^N_{i=1} \wp(x- x_i(t))$$ 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 $u(x,t)$.

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