178,586 research outputs found
The r -Hunter–Saxton equation, smooth and singular solutions and their approximation
No full textAbstract: In this work we introduce the r-Hunter–Saxton equation, a generalisation of the Hunter–Saxton equation arising as extremals of an action principle posed in L r . We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter–Saxton equation
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The r-Hunter–Saxton equation, smooth and singular solutions and their approximation
No full textIn this work we introduce the r-Hunter–Saxton equation, a generalisation of the Hunter–Saxton equation arising as extremals of an action principle posed in L r . We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter–Saxton equation
A geometric view on the generalized Proudman-Johnson and -Hunter-Saxton equations
No full textWe show that two families of equations on the real line, the generalized
inviscid Proudman--Johnson equation, and the -Hunter--Saxton equation
(recently introduced by Cotter et al.) coincide for a certain range of
parameters. This gives a new geometric interpretation of these
Proudman--Johnson equations as geodesic equations of right invariant
homogeneous -Finsler metrics on an appropriate diffeomorphism group on
. Generalizing a construction of Lenells for the Hunter--Saxton
equation, we analyze the -Hunter--Saxton equation using an isometry from the
diffeomorphism group to an appropriate subset of real-valued functions. Thereby
we show that the periodic case is equivalent to the geodesic equation on the
-sphere in the space of functions, and the non-periodic case is equivalent
to a geodesic flow on a flat space. This allows us to give explicit solutions
to these equations in the non-periodic case, and answer several questions of
Cotter et al. regarding their limiting behavior.Comment: version 3: corrected an error regarding the equivalence of the
equations in the periodic case (there is equivalence only in the non-periodic
case). Abstract and introduction were changed accordingly. version 2: minor
change
Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm
No full textWe use a three-dimensional Gerchberg–Saxton algorithm (Shabtay (2003) Opt. Commun. 226 33) to calculate the Fourier-space representation of physically realizable light beams with arbitrarily shaped three-dimensional intensity distributions. From this representation we extract a phase-hologram pattern that allows us to create such light beams experimentally. We show several examples of experimentally shaped light beams
On the Hunter-Saxton equation
No full textThe Cauchy problem for a two-component Hunter-Saxton equation, begin{align*}(u_t+uu_x)_x&=frac{1}{2}u_x^2+frac{1}{2}rho^2,rho_t+(urho)_x) &= 0,end{align*}on is studied. Conservative and dissipative weak solutions are defined and shown to exist globally. This is done by explicitly solving systems of ordinary differential equation in the Lagrangian coordinates, and using these solutions to construct semigroups of conservative and dissipative solutions
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No full texttiré de G. Cambdene, Chr. Saxton, I. Speede, T. Pont, R. Gordon, et de I. B. Boazius ; par G. Valck"Auec priuilege"Nebenkarte oben links (Färöer, Shetland und Orkney Inseln. - 17 x 17 cm)Titelkartusche oben rechts, Massstabskartusche unten link
The modified Hunter-Saxton equation
No full textGorka, P (Gorka, Przemyslaw)2,3. 2. Univ Talca, Inst Matemat & Fis, Talca, Chile
3. Warsaw Univ Technol, Dept Math & Informat Sci, PL-00661 Warsaw, PolandWe introduce a quadratic pseudo-potential for the Hunter-Saxton equation (HS), as an application of the fact that HS describes pseudo-spherical surfaces. We use it to compute conservation laws and to obtain a full Lie algebra of nonlocal symmetries for HS which contains a semidirect sum of the loop algebra over sl(2, R) and the centerless Virasoro algebra. We also explain how to find families of solutions to HS obtained using our symmetries, and we apply them to the construction of a recursion operator. We then reason by analogy with the theory of the Korteweg-de Vries and Camassa-Holm equations and we define a "modified" Hunter-Saxton (mHS) equation connected with HS via a "Miura transform". We observe that this new equation describes pseudo-spherical surfaces (and that therefore it is the integrability condition of an sl(2, R)-valued over-determined linear problem), we present two conservation laws, and we solve an initial value problem with Dirichlet boundary conditions. We also point out that our mHS equation plus its corresponding Miura transform are a formal Backlund transformation for HS. Thus, our result on existence and uniqueness of solutions really is a rigorous analytic statement on Backlund transformations. (C) 2012 Elsevier B.V. All rights reserved
Uniqueness of Dissipative Solutions for the Hunter--Saxton Equation
No full textDenne oppgaven utforsker unikheten av globale, dissipative løsninger av den symmetrisk integrerte Hunter–Saxton-ligningen med initialdata u0 ∈ E2 med derivert u0,x nesten overalt begrenset ovenfra, hvor E2 er mengden av funksjoner i H1(R) med asymptoter lagt til. Metoden er basert på allerede eksisterende bevis av unikhet for dissipative løsninger av Camassa–Holm-ligningen og for konservative løsninger av Hunter–Saxton-ligningen. Den består av å først vise at det finnes en dissipativ løsning for hver u0, ved å bruke karakteristikker, og deretter vise at enhver dissipativ løsning med initialdata u0 må sammenfalle med den konstruerte løsningen. Unikhetsbeviset presentert i denne oppgaven inneholder et steg som kanskje trenger ytterligere begrunnelse eller reformulering.
Som forberedelse til unikhetsbeviset, betraktes de følgende aspektene. Alle klassiske løsninger på I × R, hvor I er et tidsintervall, beskrives, inkludert de med uendelig energi. Det vises at man oppnår unikhet av klassiske løsninger ved å velge akselerasjonen til en karakteristikk eller ved å velge en integrert versjon. Videre defineres svake løsninger, og eksistens av dissipative løsninger vises, som nevnt i første avsnitt.This thesis investigates the uniqueness of global, dissipative solutions for the symmetrically integrated Hunter–Saxton equation with initial data u0 ∈ E2 with derivative u0,x almost everywhere bounded from above, where E2 is the set of functions in H1(R) with asymptotes added. The procedure is based on already existing proofs of uniqueness of dissipative solutions for the Camassa–Holm equation and of conservative solutions for the Hunter–Saxton equation. It consists of first showing that there exists a dissipative solution of the symmetrically integrated Hunter–Saxton equation for each u0, by using characteristics, and then showing that any dissipative solution with initial data u0 must coincide with the constructed solution. The uniqueness proof presented in this thesis contains a step that may need additional justification or reformulation.
As preparation for the uniqueness proof, the following aspects are considered. All classical solutions on I × R, where I is a time interval, are described, including those with infinite energy. It is shown that one obtains uniqueness of classical solutions by choosing the acceleration of one characteristic or by choosing an integrated version. Further, weak solutions are defined, and, as mentioned in the first paragraph, existence of dissipative solutions is established
Generic regularity and Lipschitz metric for the Hunter-Saxton type equations
No full textThe Hunter-Saxton equation determines a flow of conservative solutions taking values in the space H1(R+). However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural H1 distance. The aim of this paper is to first study the generic properties of conservative solutions of some initial boundary value problems to the Hunter-Saxton type equations. Then using these properties, we give a new way to construct a Finsler type metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H1(R+)Journal of Differential Equations 262(2), 1023-1063. (2017)0022-039
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