The authors derive all polynomial Hamiltonians of degrees 3, 4, 5 and 7 with only meromorphic solutions, producing 12 standard forms including new quartic and quintic examples for modified Painlevé equations, with none existing for degree 6 or higher.
Gambier:Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes
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The Gurevich-Pitaevskii solution to KdV, known to satisfy a self-similar reduction from the next hierarchy member, must obey a first-order PDE if any lower-order one and admits a converging Laurent series in x and t.
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Modified Painlev\'e systems with meromorphic solutions for polynomial Hamiltonians of all degrees
The authors derive all polynomial Hamiltonians of degrees 3, 4, 5 and 7 with only meromorphic solutions, producing 12 standard forms including new quartic and quintic examples for modified Painlevé equations, with none existing for degree 6 or higher.
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On the Gurevich-Pitaevskii solution of KdV
The Gurevich-Pitaevskii solution to KdV, known to satisfy a self-similar reduction from the next hierarchy member, must obey a first-order PDE if any lower-order one and admits a converging Laurent series in x and t.