The Diverse World of PDEs : Geometry and Mathematical Physics / I. S. Krasil'shchik, A. B. Sossinsky, and A. M. Verbovetsky, editors.

Format:

Book

Contributor:

Krasilʹshchik, I. S. (Iosif Semenovich), editor.

Sossinsky, A. B., editor.

Verbovetsky, Alexander, editor.

Series:

Contemporary mathematics (American Mathematical Society) ; Volume 788.

Contemporary Mathematics Series ; Volume 788

Language:

English

Subjects (All):

Differential equations, Partial--Congresses.

Differential equations, Partial.

Physical Description:

1 online resource (250 pages)

Edition:

First edition.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2023]

Summary:

This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13-17, 2021, at the Independent University of Moscow and Moscow State University, Moscow, Russia.The papers are devoted to various interrelations of nonlinear PDEs with geometry and integrable systems. The topics discussed are: gravitational and electromagnetic fields in General Relativity, nonlocal geometry of PDEs, Legendre foliated cocycles on contact manifolds, presymplectic gauge PDEs and Lagrangian BV formalism, jet geometry and high-order phase transitions, bi-Hamiltonian structures of KdV type, bundles of Weyl structures, Lax representations via twisted extensions of Lie algebras, energy functionals and normal forms of knots, and differential invariants of inviscid flows.The companion volume (Contemporary Mathematics, Volume 789) is devoted to Algebraic and Cohomological Aspects of PDEs.

Contents:

Cover

Title page

Contents

The Editors Preface

Geometric analysis of metric Legendre foliated cocycles on contact manifolds via SODE structure

1. Introduction

2. Frölicher-Nijenhuis formalism and projective metrizability problem

3. Legendre foliations on contact manifolds

4. Metric Legendre foliated cocycles compatible with SODE structure

Conclusions

References

Special Vinberg cones, invariant admissible cubics and special real manifolds

2. -algebras and associated Vinberg cones

3. Invariant rational functions of a Vinberg cone

4. Examples of -algebras and associated Vinberg cones

5. Construction of special real manifolds with large isometry groups

Acknowledgments

Bundles of Weyl structures and invariant calculus for parabolic geometries

1. Parabolic geometries and the bundle of Weyl structures

2. The invariant calculus

3. The nearly invariant operators

Coverings and pseudosymmetries of differential equations

2. Coverings of differential equations

3. Infinite jets

4. Mappings of diffieties

5. Pseudosymmetries of differential equations

6. Integrable pseudosymmetries

7. Example of calculating coverings from a control system

8. Commutation of pseudosymmetries

9. Multivector pseudosymmetries

10. Generators of modules of Cartan -forms for control systems

11. Conclusion

Differential invariants of inviscid flows in pipes

Introduction

1. The Euler system

2. Flows in pipes

Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles

2. Preliminaries

3. Local gauge theories as gauge PDEs

4. Presymplectic gauge PDEs

5. Weak presymplectic gauge PDEs

6. Induced gauge PDEs on submanifolds and boundaries.

Appendix A. Technical details

Minimal realizations of the KP hierarchy, its strict version and their reductions

2. The hierarchies in a wider perspective

3. Reduction to a standard setting

4. The minimal realizations

5. Scaling invariance

Minimal coupling of gravitational and electromagnetic fields in General Relativity

1. Spacetime

2. Gravitational and electromagnetic fields

3. Observed expressions

Projective-geometric aspects of bi-Hamiltonian structures of KdV type

3. Conditions of compatibility

4. Conclusion

Acknowledgment

Measurement of random operators, jet geometry and high-order phase transitions

2. Measurement of random operators

3. Jet geometry and probability

4. Random endomorphisms

Metrics admitting projective and c-projective vector fields

2. Projective structures, their metrizability and degree of mobility

3. Lie's classical problems

4. Higher dimensional extensions

5. Complex Lie problem

Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results

2. Preliminaries and notation

3. Lie algebras of the Kac-Moody type and their extensions

4. Equation associated to ₂⋉ _{1,-1}

5. 3D generalized Hunter-Saxton equation

6. Equation associated to ₂⋉ _{2,-1}

7. Equation associated to ⊕

8. Concluding remarks

Energy functionals and normal forms of knots and plane curves

1. Main definitions

2. Basic properties

3. Two reasonable conjectures

4. Energy functionals for thickened knots and normal forms

5. Physical experiments.

6. Circular and almost flat normal forms

7. Normal forms vs. ideal knots

Ackowledgements

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Krasil′shchik, I. S. The Diverse World of PDEs

ISBN:

1-4704-7408-5