# School of Mathematics

## Dr Vincent Caudrelier

Lecturer

Applied Mathematics

### Contact details

Room: 9.304

Tel: +44 (0)113 3439522

Email: v.caudrelier @ leeds.ac.uk

### Keywords

Classical and quantum integrable systems

Classical and quantum inverse scattering method

Soliton equations

Boundaries and defects

Integrable PDEs on graphs

Yang-Baxter and reflection equations

Poisson-Lie groups and quantum groups

## Research interests

I am interested in the area of Mathematical Physics known as integrable systems. They appear in all sorts of areas: classical and quantum mechanics, classical and quantum field theory, statistical mechanics, and in various forms: evolutionary models over discrete, semi-discrete or continuous spacetime or non-evolutionary. They share common features that are encapsulated in rich and important mathematical structures like Poisson-Lie groups, for classical integrable (field) theories and quantum groups, for quantum integrable (field) theories. The most famous equation related to these structures is the Yang-Baxter equation (classical or quantum).

They allow for exact solutions which have many applications in predicting exactly the physical behaviour of the systems they describe. For instance, correlation functions in quantum spin chains or long-time asymptotics of solutions of integrable PDEs can computed analytically and exactly. Typical domains of application are condensed matter physics, nonlinear waves dynamics in optics, fluid mechanics or plasma physics, 2D statistical models for percolation, etc.

My particular focus is on the study of the effect of boundaries and/or defects/impurities on solutions of these models. Recently, I have developed a scheme to formulate the inverse scattering method for integrable PDEs on (star) graphs.

### Useful links

## Publications

**Avan J, Caudrelier V** On the origin of dual Lax pairs and their r-matrix structure *Journal of Geometry and Physics*, **120**, 106-128, 2017

DOI:10.1016/j.geomphys.2017.05.010

View abstract

**Caudrelier V, Doyon B** The quench map in an integrable classical field theory: nonlinear Schrödinger equation *Journal of Physics A: Mathematical and Theoretical*, **49**, 2016

DOI:10.1088/1751-8113/49/44/445201

View abstract

**Avan J, Caudrelier V, Doikou A, Kundu A** Lagrangian and Hamiltonian structures in an integrable hierarchy and space–time duality *Nuclear Physics B*, **902**, 415-439, 2016

DOI:10.1016/j.nuclphysb.2015.11.024

View abstract

**Caudrelier V** On the Inverse Scattering Method for Integrable PDEs on a Star Graph *Communications in Mathematical Physics*, **338**, 893-917, 2015

DOI:10.1007/s00220-015-2378-9

View abstract

**Caudrelier V** Multisymplectic approach to integrable defects in the sine-Gordon model *Journal of Physics A: Mathematical and Theoretical*, **48**, 1-23, 2015

DOI:10.1088/1751-8113/48/19/195203

View abstract

**Caudrelier V, Kundu A** A multisymplectic approach to defects in integrable classical field theory *Journal of High Energy Physics*, **2015**

DOI:10.1007/JHEP02(2015)088

View abstract

**Caudrelier V, Mintchev M, Ragoucy E** Exact scattering matrix of graphs in magnetic field and quantum noise *Journal of Mathematical Physics*, **55**, 083524-083524, 2014

DOI:10.1063/1.4893354

View abstract

**Caudrelier V, Zhang QC** Yang-Baxter and reflection maps from vector solitons with a boundary *Nonlinearity*, **27**, 1081-1103, 2014

View abstract

**Caudrelier V, Crampé N, Zhang QC** Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency *SIGMA 10*, **014**, 2014

DOI:10.3842/SIGMA.2014.014

View abstract

**Caudrelier V, Mintchev M, Ragoucy E** Quantum wire network with magnetic flux *Physics Letters A*, 2013

**Caudrelier V, Crampe N, Zhang QC** Set-theoretical reflection equation: Classification of reflection maps *J. Phys. A: Math. Theor.*, **46**, 095203-095203, 2012

DOI:10.1088/1751-8113/46/9/095203

View abstract

**Caudrelier V, Zhang QC** Vector Nonlinear Schrödinger Equation on the half-line *J. Phys. A: Math. Theor.*, **45**, 105201-105201, 2011

DOI:10.1088/1751-8113/45/10/105201

View abstract

**Caudrelier V, Ragoucy E** Direct computation of scattering matrices for general quantum graphs *Nucl.Phys.B*, **828**, 515-535, 2009

DOI:10.1016/j.nuclphysb.2009.10.012

View abstract

**Caudrelier V, Crampe N** Symmetries of Spin Calogero Models *Symmetry, Integrability and Geometry : Methods and Applications*, **4**, 2008

**Caudrelier V** On a systematic approach to defects in classical integrable field theories *IJGMMP*, **5**, 1085-1108, 2007

DOI:10.1142/S0219887808003223

View abstract

**Caudrelier V, Crampe N** Exact energy spectrum for models with equally spaced point potentials *Nucl.Phys. B*, **738**, 351-367, 2005

DOI:10.1016/j.nuclphysb.2005.12.014

View abstract

**Caudrelier V ** *Factorization in integrable systems with impurity* 2005

**Caudrelier V, Mintchev M, Ragoucy E** Solving the quantum nonlinear Schrodinger equation with delta-type impurity *J MATH PHYS*, **46**, 2005

DOI:10.1063/1.1842353

**Caudrelier V, Crampe N** Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity *Rev.Math.Phys.*, **19**, 349-370, 2005

DOI:10.1142/S0129055X07002973

View abstract

**Caudrelier V, Mintchev M, Ragoucy E, Sorba P** Reflection-Transmission Quantum Yang-Baxter Equations *J.Phys. A*, **38**, 3431-3442, 2004

DOI:10.1088/0305-4470/38/15/013

View abstract

**Caudrelier V, Ragoucy E** Spontaneous symmetry breaking in the non-linear Schrodinger hierarchy with defect *J.Phys. A*, **38**, 2241-2258, 2004

DOI:10.1088/0305-4470/38/10/013

View abstract

**Caudrelier V, Mintchev M, Ragoucy E** The quantum non-linear Schrodinger model with point-like defect *J.Phys. A*, **37**, L367-L376, 2004

DOI:10.1088/0305-4470/37/30/L02

View abstract

**Caudrelier V, Ragoucy E** Lax pair and super-Yangian symmetry of the nonlinear super-Schrodinger equation *J MATH PHYS*, **44**, 5706-5732, 2003

DOI:10.1063/1.1625078

**Caudrelier V, Crampe N** Integrable N-particle Hamiltonians with Yangian or Reflection Algebra Symmetry *J.Phys.A*, **37**, 6285-6298, 2003

DOI:10.1088/0305-4470/37/24/007

View abstract

**Caudrelier V, Ragoucy E** Quantum resolution of the nonlinear super-Schrodinger equation *Int.J.Mod.Phys. A*, **19**, 1559-1578, 2003

DOI:10.1142/S0217751X0401804X

View abstract

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