| 1. |
- Breit, D., et al.
(författare)
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A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth
- 2019
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Ingår i: Journal of Nonlinear Science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 29:6, s. 2601-2655
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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.
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| 2. |
- Chen, R. M., et al.
(författare)
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Stability of the μ-Camassa-Holm peakons
- 2013
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Ingår i: Journal of nonlinear science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 23:1, s. 97-112
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Tidskriftsartikel (refereegranskat)abstract
- The μ-Camassa-Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa-Holm equation. We prove that the periodic peaked traveling wave solutions (peakons) of the μCH equation are orbitally stable.
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| 3. |
- Compagnoni, Marco, et al.
(författare)
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The Algebro-geometric Study of Range Maps
- 2017
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Ingår i: Journal of nonlinear science. - : SPRINGER. - 0938-8974 .- 1432-1467. ; 27:1, s. 99-157
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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.
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| 4. |
- Constantin, Adrian, et al.
(författare)
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Stability of the Camassa-Holm solitons
- 2002
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Ingår i: Journal of Nonlinear Science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 12:4, s. 415-422
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Tidskriftsartikel (refereegranskat)abstract
- We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances.
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| 5. |
- Fokas, A. S., et al.
(författare)
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Boundary value problems for the elliptic sine-gordon equation in a semi-strip
- 2013
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Ingår i: Journal of nonlinear science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 23:2, s. 241-282
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Tidskriftsartikel (refereegranskat)abstract
- We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann-Hilbert problem whose "jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann-Hilbert problem whose "jump matrix" depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.
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| 6. |
- Gustafsson, Björn, 1947-
(författare)
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Vortex Pairs and Dipoles on Closed Surfaces
- 2022
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Ingår i: Journal of nonlinear science. - : Springer Nature. - 0938-8974 .- 1432-1467. ; 32:5
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Tidskriftsartikel (refereegranskat)abstract
- We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present paper, we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section, we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces.
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| 7. |
- Gyllenberg, M., et al.
(författare)
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Bifurcation analysis of a metapopulation model with sources and sinks
- 1996
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Ingår i: Journal of nonlinear science. - 0938-8974 .- 1432-1467. ; 6:4, s. 329-366
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Tidskriftsartikel (refereegranskat)abstract
- A class of functions describing the Allee effect and local catastrophes in population dynamics is introduced and the behaviour of the resulting one-dimensional discrete dynamical system is investigated in detail. The main topic of the paper is a treatment of the two-dimensional system arising when an Allee function is coupled with a function describing the population decay in a so-called sink. New types of bifurcation phenomena are discovered and explained. The relevance of the results for metapopulation dynamics is discussed.
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| 8. |
- Kavallaris, Nikos I., et al.
(författare)
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Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain
- 2020
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Ingår i: Journal of nonlinear science. - : Springer. - 0938-8974 .- 1432-1467. ; 31:1
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Tidskriftsartikel (refereegranskat)abstract
- The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.
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| 9. |
- Lenells, Jonatan, 1981-
(författare)
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Dressing for a novel integrable generalization of the nonlinear Schrödinger equation
- 2010
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Ingår i: Journal of nonlinear science. - : Springer Science and Business Media LLC. - 0938-8974 .- 1432-1467. ; 20:6, s. 709-722
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Tidskriftsartikel (refereegranskat)abstract
- We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.
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| 10. |
- Lenells, Jonatan, 1981-, et al.
(författare)
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Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations
- 2019
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Ingår i: Journal of nonlinear science. - : Springer. - 0938-8974 .- 1432-1467. ; 29:4, s. 1621-1657
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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.
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