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Poisson structures on weak Sobolev loop spaces and applications to integrable systems

Jean-Pierre Magnot

TL;DR

The paper develops a rigorous analytic framework for Poisson geometry on weak Sobolev loop spaces $W^{s,p}$ with $0<s\le\tfrac{1}{2}$ and $1<p<\infty$, extending Mokhov's constructions from smooth loops to low-regularity settings. It establishes that the horizontal-vertical variational bicomplex persists in this setting, so skew-symmetry and the Jacobi identity remain valid, and it provides kernel and transgression representations for canonical 1-forms and presymplectic structures. Local (order-1) and weakly nonlocal Poisson operators are constructed and shown to be well posed on these spaces, with higher-order operators requiring stronger regularity. The framework is then applied to embed KdV, NLS, Camassa–Holm, and Dubrovin–Novikov hydrodynamic systems into the weak Sobolev setting, offering a cohomological, bicomplex-based viewpoint on Poisson structures and their deformations in the rough-function regime, and paving the way for extending Hamiltonian theory of integrable systems beyond smooth categories.

Abstract

We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces $W^{s,p}(\mathbb{S^1},\mathbb{R}^m)$ with $o < s \frac{1}{2}$ and $1 < p < \infty.$ Within this setting we construct presymplectic and Poisson structures of hydrodynamic type, as well as their weakly non local deformations involving inverse derivatives. The analytic backbone relies on the boundedness of fractional multipliers, Hilbert transforms, and Lipschitz Nemytski operators on $W^{s,p}$, which ensures that all operations used in Mokhov's formalisn remain well defined at this level of regularity. We further show that teh horizontal-vertical bicomplex underlying variational Poisson geometry can be extended to $W^{s,p},$ so that the cohomological arguments proving skew-symmetry and the Jacobi identity carry over verbatim. As an application, we embed the Hamiltonian formalisms of several integrable PDEs (KdV, nonlinear Schrödinger, Camassa--Holm, and hydrodynamic systems of Dubrovin--Novikov type) into this weak Sobolev setting. Local order-one Poisson operators and their weakly nonlocal extensions are shown to be well posed for $W^{s,p}$ loops, while higher-order operators (e.g. the second KdV bracket) require stronger regularity. Our results provide a rigorous analytic foundation for Poisson geometry on weak loop spaces and open the way for extending the Hamiltonian theory of integrable systems beyond the smooth category.

Poisson structures on weak Sobolev loop spaces and applications to integrable systems

TL;DR

The paper develops a rigorous analytic framework for Poisson geometry on weak Sobolev loop spaces with and , extending Mokhov's constructions from smooth loops to low-regularity settings. It establishes that the horizontal-vertical variational bicomplex persists in this setting, so skew-symmetry and the Jacobi identity remain valid, and it provides kernel and transgression representations for canonical 1-forms and presymplectic structures. Local (order-1) and weakly nonlocal Poisson operators are constructed and shown to be well posed on these spaces, with higher-order operators requiring stronger regularity. The framework is then applied to embed KdV, NLS, Camassa–Holm, and Dubrovin–Novikov hydrodynamic systems into the weak Sobolev setting, offering a cohomological, bicomplex-based viewpoint on Poisson structures and their deformations in the rough-function regime, and paving the way for extending Hamiltonian theory of integrable systems beyond smooth categories.

Abstract

We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces with and Within this setting we construct presymplectic and Poisson structures of hydrodynamic type, as well as their weakly non local deformations involving inverse derivatives. The analytic backbone relies on the boundedness of fractional multipliers, Hilbert transforms, and Lipschitz Nemytski operators on , which ensures that all operations used in Mokhov's formalisn remain well defined at this level of regularity. We further show that teh horizontal-vertical bicomplex underlying variational Poisson geometry can be extended to so that the cohomological arguments proving skew-symmetry and the Jacobi identity carry over verbatim. As an application, we embed the Hamiltonian formalisms of several integrable PDEs (KdV, nonlinear Schrödinger, Camassa--Holm, and hydrodynamic systems of Dubrovin--Novikov type) into this weak Sobolev setting. Local order-one Poisson operators and their weakly nonlocal extensions are shown to be well posed for loops, while higher-order operators (e.g. the second KdV bracket) require stronger regularity. Our results provide a rigorous analytic foundation for Poisson geometry on weak loop spaces and open the way for extending the Hamiltonian theory of integrable systems beyond the smooth category.
Paper Structure (50 sections, 19 theorems, 94 equations, 1 table)

This paper contains 50 sections, 19 theorems, 94 equations, 1 table.

Table of Contents

  1. Introduction
  2. Sobolev and BV loop spaces
  3. The Gagliardo (Slobodeckij) definition
  4. Fractional powers of the Laplacian on $\mathbb S^1$
  5. Equivalence of the two definitions
  6. Useful consequences
  7. Extension of the fundamental 1-form to weak loop spaces with values in $\mathbb R^2$
  8. Extension to $H^{1/2}$
  9. Generalization to $W^{s,p}$, $0<s\le\frac{1}{2}$, $1<p<\infty$
  10. Integral kernel representation (case $H^{1/2}$)
  11. The BV case
  12. Variable 2-forms
  13. Kernel representation of the fundamental 1-form on $H^{1/2}$
  14. A scalar bilinear identity at $s=\frac{1}{2}$
  15. Vector-valued case and the matrix $J$
  16. ...and 35 more sections

Key Result

Proposition 3

Let $0<s<1$. There exists a constant $c_{s,\mathbb S^1}\in(0,\infty)$ such that for all $u\in C^\infty(\mathbb S^1)$ with mean zero, In particular, $[u]_{W^{s,2}}\asymp \||D|^s u\|_{L^2}$ and $\|u\|_{H^s}^2=\|u\|_{L^2}^2+[u]_{W^{s,2}}^2$ with equivalent norms.

Theorems & Definitions (64)

  • Definition 1: Spectral (Bessel) definition for $p=2$
  • Definition 2: Bessel potential spaces for $1<p<\infty$
  • Proposition 3: Quadratic form identity on $\mathbb S^1$
  • proof : Sketch with references
  • Remark 4
  • Theorem 5: Equivalence of norms on $\mathbb S^1$
  • proof : Sketch with references
  • Proposition 6: Exterior differential of the transgressed 1--form
  • proof : Sketch of proof
  • Remark 7: Extension to weak loop spaces
  • ...and 54 more