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Pei, Long

Open this publication in new window or tab >>A note on well-posedness of bidirectional Whitham equation### Pei, Long

### Wang, Yuexun

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 98, p. 215-223Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

PERGAMON-ELSEVIER SCIENCE LTD, 2019
##### Keywords

Whitham-type equations, Dispersive equations, Well-posedness
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-260153 (URN)10.1016/j.aml.2019.06.015 (DOI)000483423900032 ()2-s2.0-85067842931 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway..

We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces, using a square-root type transformation to symmetrise the system.

QC 20191001

Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-01Bibliographically approvedOpen this publication in new window or tab >>Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations### Lenells, Jonatan

### Pei, Long

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Journal of nonlinear science, ISSN 0938-8974, E-ISSN 1432-1467, Vol. 29, no 4, p. 1621-1657Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2019
##### Keywords

Ernst equation, Einstein equations, Boundary value problem, Unified transform method, Fokas method, Riemann-Hilbert problem, Theta function
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-257457 (URN)10.1007/s00332-018-9527-1 (DOI)000480743200011 ()2-s2.0-85059591310 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

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.

QC 20190830

Available from: 2019-08-30 Created: 2019-08-30 Last updated: 2019-09-05Bibliographically approvedOpen this publication in new window or tab >>Symmetric solutions of evolutionary partial differential equations### Bruell, Gabriele

### Ehrnstrom, Mats

### Geyer, Anna

### Pei, Long

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 10, p. 3932-3950Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

IOP PUBLISHING LTD, 2017
##### Keywords

evolution equations, symmetry, nonlocal equations, Euler equations
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-215438 (URN)10.1088/1361-6544/aa8427 (DOI)000411158000001 ()2-s2.0-85030166337 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from EhrnstrOm et al (2009 Int. Math. Res. Not. 2009 4578-96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases.

QC 20171019

Available from: 2017-10-19 Created: 2017-10-19 Last updated: 2017-10-19Bibliographically approved