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The Painleve Test of Higher Dimensional KdV Equation

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 Added by Yu Song-Ju
 Publication date 1996
  fields Physics
and research's language is English




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We argue the integrability of the generalized KdV(GKdV) equation using the Painleve test. For $d( le 2)$ dimensional space, GKdV equation passes the Painleve test but does not for $d geq 3$ dimensional space. We also apply the Ablowitz-Ramani-Segurs conjecture to the GKdV equation in order to complement the Painleve test.



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Four 4-dimensional Painleve-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painleve system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painleve-type equations. If we only consider Painleve-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painleve-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are neatly written using Hamiltonians of the classical Painleve equations.
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