Stationary Expansion Shocks for a Regularized Boussinesq System

El, Gennady; Hoefer, Mark; Shearer, Michael

doi:10.1111/sapm.12191

Cited by 3 publications

(9 citation statements)

References 8 publications

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“…The fact that numerically resolved transition fronts sharpen as the dispersive time scale tends to zero while the limiting periodic patterns maintain their amplitude suggest weak convergence. However, similar to the examples presented in [15,16], the formal weak limits of the obtained fronts are unstable (entropically non-admissible) in the framework of the original p-system. They owe their stability exclusively to dispersive regularization and therefore, instead of the p-system, the limiting measure valued profiles [37,38,9] can be expected to serve as admissible weak solutions of the Whitham-type higher order hyperbolic system [66,12,14,57].…”

Section: Introductionsupporting

confidence: 70%

“…with (16) and (17) remaining equivalent for smooth motions. Jump discontinuities in the regularized model must respect the generalized RH relations (11) and (12) which remain the same.…”

Section: Regularized Modelmentioning

confidence: 99%

“…The three constants w, u, p, instead of two, were introduced for later convenience. We can similarly integrate the energy balance equation (16) to obtain…”

Section: Regularized Modelmentioning

confidence: 99%

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Stationary shock-like transition fronts in dispersive systems

et al. 2020

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We show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine-Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not traveling waves and apparently require for their description more complex anzats, evolve into traveling wave type jump discontinuities.

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Section: Introductionsupporting

confidence: 70%

“…with (16) and (17) remaining equivalent for smooth motions. Jump discontinuities in the regularized model must respect the generalized RH relations (11) and (12) which remain the same.…”

Section: Regularized Modelmentioning

confidence: 99%

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Stationary shock-like transition fronts in dispersive systems

et al. 2020

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“…Figure 10. Note that the expansion shock structure converges to the rarefaction solution (52) as 𝑡 → ∞.…”

Section: Expansion Shocksmentioning

confidence: 84%

“…The analysis of two‐phase linear wavetrains near a zero dispersion point is general, and the coupled NLS description of DSW implosion can be applied to the conduit and magma equations where this effect is known to occur. Expansion shocks and solitary wave shedding have been observed in the regularized Boussinesq shallow water system 52 albeit in a nonphysical, short‐wave regime. Still, it would be fair to say that many of the striking nonconvex wave regimes generated in the BBM dispersive Riemann problem await their realization in other systems.…”

Section: Discussionmentioning

confidence: 99%

Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

Congy

Hoefer

et al. 2021

Stud Appl Math

Self Cite

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Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equation ut+uux=uxxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation ut+uux+uxxx=0. The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit as t→∞ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem.

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Hamiltonian shocks

Arnold,

Camassa,

Ding

2024

Stud Appl Math

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Wave propagation in the form of fronts or kinks, a common occurrence in a wide range of physical phenomena, is studied in the context of models defined by their Hamiltonian structure. Motivated, for dispersive wave evolution equations such as a strongly nonlinear model of two‐layer internal waves in the Boussinesq limit, by the symmetric properties of a class of front‐propagating solutions, known as conjugate states or solibores, a generalized formulation based purely on the dispersionless reduction of a system is introduced, and a class of undercompressive shock solutions, here referred to as “Hamiltonian shocks,” is defined. This analysis determines whether a Hamiltonian shock, representing locally a kink for the parent dispersive equations, will interact with a sufficiently smooth background wave without inducing loss of regularity, which would take the form of a classical dispersive shock for the parent equations. This property is also related to an infinitude of conservation laws, drawing a parallel to the case of completely integrable systems.

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Stationary Expansion Shocks for a Regularized Boussinesq System

Cited by 3 publications

References 8 publications

Stationary shock-like transition fronts in dispersive systems

Stationary shock-like transition fronts in dispersive systems

Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

Hamiltonian shocks

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