On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations

Iandoli, Felice

doi:10.48550/arxiv.2202.06710

Cited by 1 publication

(2 citation statements)

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“…For further details on the Hamiltonian structure of the KdV equation and the local well-posedness of the KdV equation in low-regularity regimes we refer the reader to [4,16,22,37].…”

Section: First Integrals and Hamiltonian Formulation Of The Kdv Equationmentioning

confidence: 99%

“…While in many situations this new class of integrators allows for approximations for much rougher data, than for instance classical splitting methods [20,21], previous resonance-based approaches lack one important property: the preservation of geometric structures. This is particularly drastic in case of the KdV equation which is completely integrable, possessing infinitely many conserved quantities [10,22], an important property which we wish to capture -at least up to some degree -also on the level of the discretisation. A revolutionary step in this direction was taken by the theory of geometric numerical integration [11,17,28,36] resulting in the development of a wide range of structure-preserving algorithms firstly for dynamical systems and later also for partial differential equations with conservation laws [3,7,12,33,32].…”

Section: Introductionmentioning

confidence: 99%

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Bridging the gap: symplecticity and low regularity on the example of the KdV equation

Maierhofer¹,

Schratz²

2022

Preprint

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Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric structures. This is particularly drastic in the case of the Korteweg-de Vries (KdV) equation which is a fundamental model in the broad field of dispersive equations that is completely integrable, possessing infinitely many conserved quantities, an important property which we wish to capture -at least up to some degree -also on the discrete level. A revolutionary step in this direction was set by the theory of geometric numerical integration resulting in the development of a wide range of structure-preserving algorithms for Hamiltonian systems. However, in general, these methods rely heavily on highly regular solutions. State-of-theart low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. This work makes a first step towards bridging the gap between low regularity and structure preservation. We introduce a novel symplectic (in the Hamiltonian picture) resonancebased method on the example of the KdV equation that allows for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level.

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“…For further details on the Hamiltonian structure of the KdV equation and the local well-posedness of the KdV equation in low-regularity regimes we refer the reader to [4,16,22,37].…”

Section: First Integrals and Hamiltonian Formulation Of The Kdv Equationmentioning

confidence: 99%