arXiv subjects
Publications and source records attributed to Jonatan Lenells.
We consider the rigorous derivation of asymptotic formulas for initial-boundary value problems using the nonlinear steepest descent method. We give detailed derivations of the asymptotics in the similarity and self-similar sectors for the mKdV equation in the quarter plane. Precise and uniform error estimates are presented in detail.
We consider solutions of the defocusing nonlinear Schrödinger equation in the quarter plane whose Dirichlet boundary data approach a single exponential $αe^{iωt}$ as $t \to \infty$. In order to determine the long time asymptotics of the solution, it is necessary to first characterize the asymptotic behavior of the Neumann value in terms of the given data. Assuming that the initial data decay as $x \to \infty$, we derive necessary conditions for the Neumann value to asymptote towards a single exponential of the form $ce^{iωt}$. Since our approach yields expressions which relate $α$, $ω$, and $c$, the result can be viewed as a characterization of the large $t$ behavior of the Dirichlet to Neumann map for single exponential profiles.
We study gravitational collapse of a spherical fluid in nonrelativistic general covariant theory of the Hořava-Lifshitz gravity with the projectability condition and an arbitrary coupling constant $λ$, where $|λ- 1|$ characterizes the deviation of the theory from general relativity in the infrared limit. The junction conditions across the surface of a collapsing star are derived under the (minimal) assumption that the junctions be mathematically meaningful in terms of distribution theory. When the collapsing star is made of a homogeneous and isotropic perfect fluid, and the external region is described by a stationary spacetime, the problem reduces to the matching of six independent conditions. If the perfect fluid is pressureless (a dust fluid), it is found that the matching is also possible. In particular, in the case $λ= 1$, the external spacetime is described by the Schwarzschild (anti-) de Sitter solution written in Painlevé-Gullstrand coordinates. In the case $λ\not= 1$, the external spacetime is static but not asymptotically flat. Our treatment can be easily generalized to other versions of Hořava-Lifshitz gravity or, more generally, to any theory of higher-order derivative gravity.
The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann-Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of $L^2$-RH problems, we consider the construction of quarter plane solutions which are $C^1$ in time and $C^3$ in space.
It has been conjectured that the defocusing nonlinear Schrödinger (NLS) equation on the half-line does not admit solitons. We give a proof of this conjecture.
We consider solutions of the defocusing nonlinear Schrödinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions, characterizes the pairs of periodic functions which can arise as Dirichlet and Neumann values for large $t$ in this way. The theorem also provides a constructive way of determining explicit solutions with the given periodic boundary values. Hence our approach leads to a class of new exact solutions of the defocusing NLS equation on the half-line.
We develop a theory of $n \times n$-matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour $Γ$ is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of $L^p$-Riemann-Hilbert problem and establish basic uniqueness results and a vanishing lemma. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.
We consider initial-boundary value problems for the KdV equation $u_t + u_x + 6uu_x + u_{xxx} = 0$ on the half-line $x \geq 0$. For a well-posed problem, the initial data $u(x,0)$ as well as one of the three boundary values $\{u(0,t), u_x(0,t), u_{xx}(0,t)\}$ can be prescribed; the other two boundary values remain unknown. We provide a characterization of the unknown boundary values for the Dirichlet as well as the two Neumann problems in terms of a system of nonlinear integral equations. The characterizations are effective in the sense that the integral equations can be solved perturbatively to all orders in a well-defined recursive scheme.
Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter-Saxton (2HS) system, that displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifold which is isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulas for the solutions of 2HS. We also show that when restricted to functions of zero mean, 2HS reduces to the geodesic equation on an infinite-dimensional manifold which admits a Kähler structure. We demonstrate that this manifold is in fact isometric to a subset of complex projective space, and that the above constructions provide an example of an infinite-dimensional Hopf fibration.
We study the family of $α$-connections of Amari-Chentsov on the homogeneous space $\mathcal{D}(M)/\mathcal{D}_μ(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.
We analyze a class of initial-boundary value problems for the Degasperis-Procesi equation on the half-line. Assuming that the solution $u(x,t)$ exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $k$.
We show that the weakly dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton, and Novikov equations can be reduced to their non-dissipative versions by means of an exponentially time-dependent scaling. Hence, up to a simple change of variables, the non-dissipative and dissipative versions of these equations are equivalent. Similar results hold also for the equations in the so-called b-family of equations as well as for the two-component and μ-versions of the above equations.
We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an $\mathfrak{sl}(3)$ Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method are presented.
We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve $3 \times 3$ matrices. Whereas initial value problems for integrable equations can be analyzed by means of the classical Inverse Scattering Transform (IST), the presence of a boundary presents new challenges. Over the last fifteen years, an extension of the IST formalism developed by Fokas and his collaborators has been successful in analyzing boundary value problems for several of the most important integrable equations with $2 \times 2$ Lax pairs, such as the Korteweg-de Vries, the nonlinear Schrödinger, and the sine-Gordon equations. In this paper, we extend these ideas to the case of equations with Lax pairs involving $3 \times 3$ matrices.
We show that the two-component Hunter-Saxton system with negative coupling constant describes the geodesic flow on an infinite-dimensional pseudosphere. This approach yields explicit solution formulae for the Hunter-Saxton system. Using this geometric intuition, we conclude by constructing global weak solutions. The main novelty compared with similar previous studies is that the metric is indefinite.
We systematically study black holes in the Horava-Lifshitz (HL) theory by following the kinematic approach, in which a horizon is defined as the surface at which massless test particles are infinitely redshifted. Because of the nonrelativistic dispersion relations, the speed of light is unlimited, and test particles do not follow geodesics. As a result, there are significant differences in causal structures and black holes between general relativity (GR) and the HL theory. In particular, the horizon radii generically depend on the energies of test particles. Applying them to the spherical static vacuum solutions found recently in the nonrelativistic general covariant theory of gravity, we find that, for test particles with sufficiently high energy, the radius of the horizon can be made as small as desired, although the singularities can be seen in principle only by observers with infinitely high energy. In these studies, we pay particular attention to the global structure of the solutions, and find that, because of the foliation-preserving-diffeomorphism symmetry, ${Diff}(M,{\cal{F}})$, they are quite different from the corresponding ones given in GR, even though the solutions are the same. In particular, the ${Diff}(M,{\cal{F}})$ does not allow Penrose diagrams. Among the vacuum solutions, some give rise to the structure of the Einstein-Rosen bridge, in which two asymptotically flat regions are connected by a throat with a finite non-zero radius. We also study slowly rotating solutions in such a setup, and obtain all the solutions characterized by an arbitrary function $A_{0}(r)$. The case $A_{0} = 0$ reduces to the slowly rotating Kerr solution obtained in GR.
We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.
Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev $H^0$ and $H^1$ energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs.