Traveling Wave Solutions of the Modified Korteweg-deVries-Burgers Equation

Jacobs, D. A. H.; Mckinney, B.; Shearer, Michael

doi:10.1006/jdeq.1995.1043

Cited by 134 publications

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“…Traveling-wave profiles of both Lax and undercompressive shock waves for the equation (1.5) have been proved to exist [12,11] and behave analogously in the limit ε → 0. We refrain from making a similar complete traveling-wave analysis for (1.9) because the current paper focuses on undercompressive waves.…”

Section: Undercompressive Shock Waves and Dissipative Admissibilitymentioning

confidence: 95%

“…Concerning (i), in [12] it is proved that if u − > 2 √ 2 3 √ γ then there is a saddle to saddle connection to…”

Section: Traveling Waves For the Diffusive-dispersive Regularizationmentioning

confidence: 99%

“…A complete study of the traveling waves in the case f (u) = u 3 , including (1.2) under the condition (1.4), can be found in [12]. Moreover, the analysis in [12] rules out the possibility of the doubly undercompressive case quoted above. Let f be chosen such that f is nonnegative or bounded.…”

Section: Introductionmentioning

confidence: 97%

“…The major interest in undercompressive waves stems from the fact that they appear in several applications as bulk interfaces representing, e.g., phase boundaries [12], saturation fronts [29], precursors in thin film flow [2] and so on. In this framework the scalar case (1.1) considered in this paper must be understood as a simplified model which, however, captures main features of the problem.…”

Section: Introductionmentioning

confidence: 99%

“…For the analysis of the sharp-interface limit ε → 0 of (1.5) we refer to [24,11]. A complete study of the traveling waves in the case f (u) = u 3 , including (1.2) under the condition (1.4), can be found in [12]. Moreover, the analysis in [12] rules out the possibility of the doubly undercompressive case quoted above.…”

Section: Introductionmentioning

confidence: 99%

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Singular limits for a parabolic–elliptic regularization of scalar conservation laws

Corli

Rohde

2012

Journal of Differential Equations

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We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial-value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic-elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive-dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic-elliptic system can be understood as a low-order approximation of the third-order diffusive-dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic-elliptic system completes the paper.

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Section: Undercompressive Shock Waves and Dissipative Admissibilitymentioning

confidence: 95%

“…Concerning (i), in [12] it is proved that if u − > 2 √ 2 3 √ γ then there is a saddle to saddle connection to…”

Section: Traveling Waves For the Diffusive-dispersive Regularizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Singular limits for a parabolic–elliptic regularization of scalar conservation laws

Corli

Rohde

2012

Journal of Differential Equations

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Undercompressive shock waves in Hall‐magnetohydrodynamics

Shukla

Sharma

2018

Math Methods in App Sciences

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Funding informationUniversity Grants Commission (UGC), India JEL Classification: 35C20; 35G25; 35L67; 76N15; 76W05We study the propagation of nonlinear waves in a Hall-magnetohydrodynamic model. An asymptotic method is used to derive the Gardner-Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second-and third-order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splitting-a phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave-fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.

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Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux

Minbashian

Giesselmann

2021

Proc Appl Math and Mech

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In this work, we investigate the capabilities of deep neural networks for solving hyperbolic conservation laws with non-convex flux functions. The behaviour of solutions to these problems depends on the underlying small-scale regularization. In many applications concerning phase transition phenomena, the regularization terms consist of diffusion and dispersion which are kept in balance in the limit. This may lead to the development of both classical and non-classical (or undercompressive) shock waves at the same time which makes the development of approximation schemes that converge towards the appropriate weak solution of these problems challenging. Here, we consider a scalar conservation law with cubic flux function as a toy model and present preliminary results of an ongoing work to study the capabilities of a deep learning algorithm called PINNs proposed in [1] for solving this problem. It consists of a feed-forward network with a hyperbolic tangent activation function along with an additional layer to enforce the differential equation.

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Traveling Wave Solutions of the Modified Korteweg-deVries-Burgers Equation

Cited by 134 publications

References 0 publications

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

Undercompressive shock waves in Hall‐magnetohydrodynamics

Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux

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