On numerical inverse scattering for the Korteweg–de Vries equation with discontinuous step-like data

Bilman, Deniz; Trogdon, Thomas

doi:10.1088/1361-6544/ab6c37

Cited by 11 publications

(12 citation statements)

References 39 publications

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“…It turns out that the BBM dispersive Riemann problem exhibits a number of features that are markedly different from the KdV dispersive Riemann problem, i.e., are nonclassical. To elucidate them, we first note that the consideration of discontinuous Riemann data () for a dispersive equation leads to anomalous features, such as the generation of waves with unbounded phase and group velocities in the KdV equation 5 . One way to avoid this unphysical behavior, is to introduce smoothed Riemann data, e.g.,

u false(x, 0 false) = \frac{u_{+} - u_{-}}{2} \tanh (\frac{x}{ξ}) + \frac{u_{+} + u_{-}}{2},

where the parameter

ξ

represents the characteristic width of the initial transition.…”

Section: Introductionmentioning

confidence: 99%

“…To elucidate them, we first note that the consideration of discontinuous Riemann data (6) for a dispersive equation leads to anomalous features, such as the generation of waves with unbounded phase and group velocities in the KdV equation. 5 One way to avoid this unphysical behavior, is to introduce smoothed Riemann data, e.g., 𝑢(𝑥, 0) = 𝑢 + − 𝑢 − 2 tanh…”

Section: Introductionmentioning

confidence: 99%

“…The small parameter 𝜀 cannot be scaled out of the equation while maintaining its asymptotic validity. For example, the transformation (5) breaks the asymptotic long-wave assumption so that the dispersive Riemann problem (4), (7) is physically relevant within the shallow water wave context only if 𝜉 ≳ 1∕ √ 𝜀 ≫ 1 and 𝑢 ± = 1 + (𝜀). In this regime, the dispersive Riemann problem (4), (7) for the BBM equation is asymptotically equivalent to the problem (3), (7) for the KdV equation.…”

Section: Introductionmentioning

confidence: 99%

“…An example scenario is the circular, free interface between two viscous, Stokes fluids whose long-wave evolution is well-approximated by the conduit equation 𝑢 𝑡 + 𝑢𝑢 𝑥 − 𝑢𝑢 txx + 𝑢 𝑡 𝑢 xx = 0. 7,8 Other examples include a class of models for magma transport in the upper mantle 𝑢 𝑡 + (𝑢 𝑛 ) 𝑥 − (𝑢 𝑛 (𝑢 −𝑚 𝑢 𝑡 ) 𝑥 ) 𝑥 = 0, where 𝑛 ∈ [2,5], 𝑚 ∈ [0, 1] are parameters 9,10 and channelized water flow beneath glaciers 𝑢 𝑡 + (𝑢 𝛼 (1 − (sgn(𝑢 𝑡 )|𝑢 𝑡 𝑢 −1 | 1∕𝑛 ) 𝑥 ) 𝛽 ) 𝑥 = 0, where 𝑛 ≥ 1, 𝛼 > 1, 𝛽 > 0 are parameters. 11 The BBM equation ( 4) captures the quadratic hydrodynamic flux (…”

Section: Introductionmentioning

confidence: 99%

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Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

Congy

Hoefer

et al. 2021

Stud Appl Math

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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.

show abstract

u false(x, 0 false) = \frac{u_{+} - u_{-}}{2} \tanh (\frac{x}{ξ}) + \frac{u_{+} + u_{-}}{2},

where the parameter

ξ

represents the characteristic width of the initial transition.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

Congy

Hoefer

et al. 2021

Stud Appl Math

View full text Add to dashboard Cite

show abstract

“…It turns out that the BBM dispersive Riemann problem exhibits a number of features that are markedly different from the KdV dispersive Riemann problem, i.e., are nonclassical. To elucidate them, we first note that the consideration of discontinuous Riemann data (1.6) for a dispersive equation leads to anomalous features, such as the generation of waves with unbounded phase and group velocities in the KdV equation [6]. One way to avoid this unphysical behavior, is to introduce smoothed Riemann data, e.g.,…”

Section: Introductionmentioning

confidence: 99%

Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

Congy,

El,

Hoefer

et al. 2020

Preprint

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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.

show abstract

An artificially-damped Fourier method for dispersive evolution equations

Liu

Trogdon

2023

Applied Numerical Mathematics

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On numerical inverse scattering for the Korteweg–de Vries equation with discontinuous step-like data

Cited by 11 publications

References 39 publications

Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

An artificially-damped Fourier method for dispersive evolution equations

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