| 1. |
- Constantin, Adrian, et al.
(författare)
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Integrability of invariant metrics on the Virasoro group
- 2006
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Ingår i: Physics Letters A. - : Elsevier BV. - 0375-9601 .- 1873-2429. ; 350:1-2, s. 75-80
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Tidskriftsartikel (refereegranskat)abstract
- Each right-invariant metric on the Virasoro group induces a Hamiltonian vector field on the dual of the Lie algebra vir equipped with the canonical Lie-Poisson structure. We show that the Hamiltonian vector fields Xk induced by the metrics given at the identity by the Hk Sobolev inner products, k≥0, are bi-Hamiltonian relative to a modified Lie-Poisson structure only for k=0 and k=1.
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| 2. |
- Dimofte, T., et al.
(författare)
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Exact results for perturbative Chern-Simons theory with complex gauge group
- 2009
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Ingår i: Communications in Number Theory and Physics. - : International Press of Boston. - 1931-4523 .- 1931-4531. ; 3:2, s. 363-443
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Tidskriftsartikel (refereegranskat)abstract
- We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group GC, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative GC invariants turn out to be interesting topological invariants, which are very different from the finite-type (Vassiliev) invariants usually studied in a theory with compact gauge group G and a trivial flat connection. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic topological quantum field theory" and conjecture (with supporting numerical evidence) that SL(2, C) Chern-Simons theory is an example of such a theory.
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| 3. |
- Kalisch, Henrik, et al.
(författare)
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Numerical study of traveling-wave solutions for the Camassa-Holm equation
- 2005
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Ingår i: Chaos, Solitons & Fractals. - : Elsevier BV. - 0960-0779 .- 1873-2887. ; 25:2, s. 287-298
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Tidskriftsartikel (refereegranskat)abstract
- We explore numerically different aspects of periodic traveling-wave solutions of the Camassa-Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied.
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| 4. |
- Khesin, B., et al.
(författare)
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Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
- 2008
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 342:3, s. 617-656
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Tidskriftsartikel (refereegranskat)abstract
- We study an equation lying 'mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.
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| 5. |
- Lenells, Jonatan, 1981-
(författare)
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A bi-Hamiltonian supersymmetric geodesic equation
- 2008
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Ingår i: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 85:1, s. 55-63
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Tidskriftsartikel (refereegranskat)abstract
- A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.
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| 6. |
- Lenells, Jonatan, 1981-, et al.
(författare)
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An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons
- 2009
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 25:11
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Tidskriftsartikel (refereegranskat)abstract
- We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schrödinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.
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| 7. |
- Lenells, Jonatan, 1981-
(författare)
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Classification of all travelling-wave solutions for some nonlinear dispersive equations
- 2007
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Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 365:1858, s. 2291-2298
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Tidskriftsartikel (refereegranskat)abstract
- We present a method for the classification of all weak travelling-wave solutions for some dispersive nonlinear wave equations. When applied to the Camassa-Holm or the Degasperis-Procesi equation, the approach shows the existence of not only smooth, peaked and cusped travelling-wave solutions, but also more exotic solutions with fractallike wave profiles.
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| 8. |
- Lenells, Jonatan, 1981-
(författare)
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Classification of traveling waves for a class of nonlinear wave equations
- 2006
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Ingår i: Journal of Dynamics and Differential Equations. - : Springer Science and Business Media LLC. - 1040-7294 .- 1572-9222. ; 18:2, s. 381-391
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Tidskriftsartikel (refereegranskat)abstract
- We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.
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| 9. |
- Lenells, Jonatan, 1981-
(författare)
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Conservation laws of the Camassa-Holm equation
- 2005
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Ingår i: Journal of Physics A. - : IOP Publishing. - 0305-4470 .- 1361-6447. ; 38:4, s. 869-880
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Tidskriftsartikel (refereegranskat)abstract
- We use the bi-Hamiltonian structure of the Camassa-Holm equation to show that its conservation laws Hn[m] are homogeneous with respect to the scaling m → λm. Moreover, a direct argument is presented proving that H-1, H-2,..., are of local character. Finally, simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the Hns.
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| 10. |
- Lenells, Jonatan, 1981-
(författare)
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Exactly solvable model for nonlinear pulse propagation in optical fibers
- 2009
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 123:2, s. 215-232
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Tidskriftsartikel (refereegranskat)abstract
- The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in 1. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.
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