| 1. |
- Breit, D., et al.
(författare)
-
A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth
- 2019
-
Ingår i: Journal of Nonlinear Science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 29:6, s. 2601-2655
-
Tidskriftsartikel (refereegranskat)abstract
- We present an existence and stability theory for gravity–capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E subject to the constraint I= 2 μ, where I is the wave momentum and 0 < μ< μ, where μ is chosen small enough for the validity of our calculations. Since E and I are both conserved quantities a standard argument asserts the stability of the set Dμ of minimisers: solutions starting near Dμ remain close to Dμ in a suitably defined energy space over their interval of existence. The solitary waves which we construct are of small amplitude and are to leading order described by the cubic nonlinear Schrödinger equation. They exist in a parameter region in which the ‘slow’ branch of the dispersion relation has a strict non-degenerate global minimum and the corresponding nonlinear Schrödinger equation is of focussing type. The waves detected by our variational method converge (after an appropriate rescaling) to solutions of the model equation as μ↓ 0.
|
|
| 2. |
- Compagnoni, Marco, et al.
(författare)
-
The Algebro-geometric Study of Range Maps
- 2017
-
Ingår i: Journal of nonlinear science. - : SPRINGER. - 0938-8974 .- 1432-1467. ; 27:1, s. 99-157
-
Tidskriftsartikel (refereegranskat)abstract
- Localizing a radiant source is a problem of great interest to many scientific and technological research areas. Localization based on range measurements is at the core of technologies such as radar, sonar and wireless sensor networks. In this manuscript, we offer an in-depth study of the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummers and Cayleys surfaces. Our work also gives new insights into the localization based on range differences.
|
|
| 3. |
- Lenells, Jonatan, 1981-, et al.
(författare)
-
Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations
- 2019
-
Ingår i: Journal of nonlinear science. - : Springer. - 0938-8974 .- 1432-1467. ; 29:4, s. 1621-1657
-
Tidskriftsartikel (refereegranskat)abstract
- For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a Dirichlet boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a Neumann boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.
|
|
| 4. |
- McLachlan, R. I., et al.
(författare)
-
Geometry of Discrete-Time Spin Systems
- 2016
-
Ingår i: Journal of Nonlinear Science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 26:5, s. 1507-1523
-
Tidskriftsartikel (refereegranskat)abstract
- Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space . In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kahler geometry, this provides another geometric proof of symplecticity.
|
|
