Bridging the gap: symplecticity and low regularity on the example of the KdV equation
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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-the-art 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) resonance-based 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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