Simple waves do not avoid eigenvalue crossings

Chumakova, Lyuba; Tabak, Esteban G.

doi:10.1002/cpa.20288

Cited by 6 publications

(16 citation statements)

References 6 publications

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“…Simple waves are of crucial importance in the study of nonlinear first‐order hyperbolic PDEs. In two‐dimensional systems, they define invariant regions due to the property that simple waves do not allow a general solution to cross it tangentially . Furthermore, for mixed‐type first‐order PDE systems, if an initial condition can be bounded by simple waves that do not themselves reach the boundary of the hyperbolic region, then the solution will remain hyperbolic until breaking.…”

Section: Results On Three‐layer Flowsmentioning

confidence: 99%

“…The set (25) to (29) consists of a closed system of seven equations for the seven unknowns ℎ 1 , ℎ 2 , ℎ 3 , 1 , 2 , 3 , and P. In solving for the pressure later, we will see that in most cases the equation for P has an elliptic nature with important consequences.…”

Section: The Shallow Water Limitmentioning

confidence: 99%

“…written above in the flow variables ℎ , . Note that the momentum equations (25) to (27) implies that…”

Section: Boundary Conditions and The Benjamin Paradoxmentioning

confidence: 99%

“…In twodimensional systems, they define invariant regions 12,24 due to the property that simple waves do not allow a general solution to cross it tangentially. 25 Furthermore, for mixed-type first-order PDE systems, if an initial condition can be bounded by simple waves that do not themselves reach the boundary of the hyperbolic region, then the solution will remain hyperbolic until breaking. Therefore, using simple waves one can build the largest such region, which can be seen as a sharp bound to on hyperbolic initial F I G U R E 9 Evolution of the mode 2 simple wave solution shown in Figure 7, now in the modal variables of (49) to (52).…”

Section: Simple Wavesmentioning

confidence: 99%

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Three‐layer flows in the shallow water limit

Viríssimo

Milewski

2019

Stud Appl Math

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We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system.

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Section: Results On Three‐layer Flowsmentioning

confidence: 99%

Section: The Shallow Water Limitmentioning

confidence: 99%

“…written above in the flow variables ℎ , . Note that the momentum equations (25) to (27) implies that…”

Section: Boundary Conditions and The Benjamin Paradoxmentioning

confidence: 99%

Section: Simple Wavesmentioning

confidence: 99%

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Three‐layer flows in the shallow water limit

Viríssimo

Milewski

2019

Stud Appl Math

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“…In seismology, perhaps one of the earliest explicit references to osculation problems may be found in a book by Levshin [7]. Then again, judging by the recent increase in the number of papers on the topic of avoided crossings, it would seem that the interest of mathematicians & physicists [8][9][10][11][12], engineering scientists [13][14][15][16] and seismologists [17][18][19][20] may have been reawakened.…”

Section: Introductionmentioning

confidence: 98%

Osculations of spectral lines in a layered medium

Kausel

Malischewsky²,

Barbosa

2015

Wave Motion

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Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

et al. 2018

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Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results. arXiv:1812.02963v1 [physics.flu-dyn]

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Simple waves do not avoid eigenvalue crossings

Abstract: General one-parameter families of matrices avoid eigenvalue crossings. It is shown that the matrices associated with the simple waves of nonlinear systems of conservation laws do not obey this rule: a subset of simple waves with nonzero measure has crossing eigenvalues.

Cited by 6 publications

References 6 publications

Three‐layer flows in the shallow water limit

Three‐layer flows in the shallow water limit

Osculations of spectral lines in a layered medium

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

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