3,420 research outputs found

    The Even and Odd Supersymmetric Hunter - Saxton and Liouville Equations

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    It is shown that two different supersymmetric extensions of the Harry Dym equation lead to two different negative hierarchies of the supersymmetric integrable equations. While the first one yields the known even supersymmetric Hunter - Saxton equation, the second one is a new odd supersymmetric Hunter - Saxton equation. It is further proved that these two supersymmetric extensions of the Hunter - Saxton equation are reciprocally transformed to two different supersymmetric extensions of the Liouville equation.Comment: typos corrected and references added. To appear in Phys.Lett

    A class of equations with peakon and pulson solutions (with an Appendix by Harry Braden and John Byatt-Smith)

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    We consider a family of integro-differential equations depending upon a parameter bb as well as a symmetric integral kernel g(x)g(x). When b=2b=2 and gg is the peakon kernel (i.e. g(x)=exp(x)g(x)=\exp(-|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained from the peakon kernel with b=3b=3. Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable. However,for b=2b=2 the family restricts to the pulson family of Fringer & Holm, which is Hamiltonian and numerically displays elastic scattering of pulses. On the other hand, for arbitrary bb it is still possible to construct a nonlocal Hamiltonian structure provided that gg is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of gg. The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon dynamical system, for any value of b1b\neq 1.Comment: Contribution to volume of Journal of Nonlinear Mathematical Physics in honour of Francesco Caloger

    On a Lagrangian reduction and a deformation of completely integrable systems

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    We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm H1H^1 in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them we found two important equations, the Camassa-Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation
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