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Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

G. A. El, M. A. Hoefer, M. Shearer

arXiv:1501.01681v3nlin.PSphysics.flu-dyn

TL;DR

This review asks how dispersive and diffusive-dispersive regularizations compare for non-convex conservation laws. It compares their Riemann-problem solutions and finds that the zero-diffusion correspondence is multivalued, including kink and contact-DSW counterparts to non-classical shock structures.

  • Problem

    For non-convex fluxes, what are the purely dispersive counterparts of undercompressive shocks and composite shock-rarefaction waves?

    Source: §5 · p. 28

  • Method

    The review compares Riemann-problem solutions of the non-convex mKdV and mKdVB equations and their zero-diffusion limits.

    Source: §7 · p. 48

  • Results

    Unlike the convex-flux case, the mKdVB-to-mKdV mapping is multivalued: undercompressive shocks correspond to kinks, while shock-rarefactions have contact-DSW counterparts.

    Source: §6.2 · p. 41, §7 · pp. 48–49

  • Takeaways & Limitations

    The comparison links hyperbolic conservation laws with nonlinear dispersive wavetrains in universal asymptotic models of diffusive-dispersive Eulerian hydrodynamics.

    Source: §7 · p. 48

    Modulation theory is long-time and formal, misses initial DSW development, and assumes local periodicity that can fail near DSW edges.

    Source: §1.2 · p. 8

Abstract

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We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

1. Introduction.

The review compares purely dispersive mKdV and diffusive-dispersive mKdVB regularizations of non-convex conservation laws, focusing on their Riemann structures and the role of dispersion sign and Whitham non-strict hyperbolicity in shaping non-classical waves and admissibility. It treats dispersive theory as a singular limit of diffusive-dispersive theory and aims to bridge nonlinear dispersive waves and hyperbolic conservation laws.

1. Introduction.

  • The review links two regularization approaches for non-convex flux, using the comparison to identify dispersive theory as a singular limit of diffusive-dispersive theory and bridge two areas of applied mathematics.It focuses on mKdVB, which is purely dispersive when ν = 0 and diffusive-dispersive when ν > 0, and presents it as a universal model under a criticality condition.Source: §1 · p. 2

1.1. Scalar conservation laws.

  • For convex flux, purely dispersive and mixed dispersive-diffusive regularizations yield different shock approximations, while non-convex flux brings both deeper differences and structural correspondences.Non-convex flux also permits composite waves and undercompressive shocks that do not satisfy the Lax entropy condition.Source: §1 · p. 2, §1.1 · p. 4

1.2. DSWs and modulation equations.

  • Whitham averaging models DSWs as slowly modulated periodic waves, with Gurevich-Pitaevskii matching yielding an expanding modulation solution for the KdV step problem.The modulation description is formal, assumes a locally periodic DSW, and has long-time asymptotic character, so it does not resolve initial development or all edge structure.Source: §1.2 · pp. 4, 7, 8

1.3. Overview of this work.

  • The review classifies long-time Riemann-problem patterns for comparable mKdV and mKdVB data by jump states and dispersion sign, finding both similarities and differences.Purely dispersive solutions typically involve expanding DSW wavetrains, whereas diffusive-dispersive solutions involve traveling waves approximating shock waves.Source: §1.3 · p. 8
  • For µ > 0, mKdV and mKdVB have undercompressive traveling waves; the mKdV versions are kinks and violate the Lax entropy condition.These scalar shocks are subsonic on both sides of the wave.Source: §1.3 · p. 9
  • For µ < 0, mKdVB shock-rarefactions limit to contact discontinuities, whose dispersive counterparts are contact DSWs; constructed these alongside KdV-type DSWs.The contact DSWs are identified with non-standard sinusoidal undular bores.Source: §1.3 · p. 9
  • The mKdVB derivation identifies conditions for its dynamics to dominate near loss of genuine nonlinearity, and the derived dispersion-coefficient sign governs Riemann-solution structure.The coefficient is expressed in terms of the original Eulerian system's pressure law and linear dispersion relation.Source: §1.3 · p. 10, §1 · p. 13
  • KdV DSWs expand with two distinct speeds, unlike steady KdVB traveling-wave shocks; KdV DSW admissibility matches the Lax entropy condition, though Rankine-Hugoniot does not apply to expanding DSWs.The review also compares traveling waves approximating classical and undercompressive shocks across oscillatory and non-oscillatory regimes.Source: §1.3 · p. 10
  • mKdV classical DSW admissibility differs from the zero-diffusion mKdVB shock conditions, while non-strict hyperbolicity enables non-classical DSWs such as kinks and contact DSWs.The admissibility conditions also delineate initial-data regions with qualitatively different solution behavior.Source: §1.3 · pp. 10–11
  • With non-convex dispersion, µ may change sign over the wave-number range involved, adding structural complexity, although such changes likely occur at short waves and may not affect the long-wave approximation.The review uses direct numerical solutions with sufficiently fine space-time discretization to resolve small-scale oscillations.Source: §1.3 · p. 10, §1 · p. 14

3. Lax shocks and their dispersive counterparts.

This section examines Lax shocks and their dispersive counterparts, comparing positive-diffusion KdVB traveling waves with expanding KdV dispersive shock waves and their Riemann-problem outcomes. It characterizes admissible mKdVB shocks and how Riemann solutions depend on the sign of μ and the dispersive regime, including one-sided contact shocks linking shock and rarefaction structures.

3. Lax shocks and their dispersive counterparts.

  • KdVB traveling waves connect the shock states if and only if the shock satisfies the Lax entropy condition.For the stated orientation, this requires u+ < u−.Source: §3 · p. 14, §3.1 · p. 15

3.1. Traveling wave solutions approximating shocks.

  • As diffusion weakens, KdVB shock profiles change from monotone to oscillatory, but at zero diffusion the connecting traveling wave disappears and a KdV solitary wave remains.For small positive diffusion, the oscillatory profile’s leading amplitude is approximately 3Δ/2.Source: §3.1 · p. 15
  • For KdVB Riemann data, u− > u+ approaches a traveling wave, while u− < u+ produces an approximately classical rarefaction wave.Dissipation and dispersion smooth the rarefaction’s corner irregularities.Source: §3.1 · p. 16

3.2. KdV equation and DSWs.

  • Changing KdV dispersion sign changes DSW orientation and solitary-wave polarity, while compressive rarefaction data retain a classical fan with one corner smoothed by dispersive oscillations.The oscillation amplitude decays like t^-1/2, and the other corner is exponentially resolved to a constant.Source: §3.2 · pp. 20–21
  • Unlike the unsteady, expanding KdV DSW with leading- and trailing-edge speeds, the KdVB traveling wave for a diffusively modified shock has one Rankine–Hugoniot speed; leading amplitudes are 2Δ and 3Δ/2, respectively, for small ν > 0.Source: §3.2 · p. 21
  • A shock with u+ = −u−/2 is characteristic on the right and is called a one-sided contact discontinuity.For the KdV shock, this satisfies the Lax entropy condition in a limiting sense.Source: §4 · p. 22

4.1. Traveling wave solutions.

  • For μ < 0, all Lax shocks are admissible and no other shocks are; the traveling-wave ODE also preserves admissibility under simultaneous sign reversal of u±.The ODE’s symmetry maps each admissible shock to one with both end states negated and the same speed.Source: §4.1 · p. 23
  • For μ > 0, phase-plane connections can admit undercompressive shocks while excluding some Lax shocks; when |u−| is small enough, no undercompressive shock occurs and all Lax shocks are admissible.At u+ = −u−/2, the one-sided contact shock is admissible for μ < 0 or, for μ > 0, when the stated bound on |u−| holds.Source: §4.1 · pp. 23–24

4.2. The Riemann problem.

  • A shock-rarefaction composite attaches a faster rarefaction beyond the one-sided contact shock, whose speed is characteristic on the right.The rarefaction’s trailing edge is discontinuous.Source: §4.2 · p. 25
  • For μ < 0, Riemann solutions use constants, rarefactions, and admissible Lax shocks; for μ > 0, undercompressive shocks and two-wave structures separated by a constant state also occur.The μ > 0 classification depends on δ, whereas the μ < 0 classification is independent of δ > 0; as μ → 0, the classifications coincide.Source: §4.2 · pp. 24, 25, 27, Figure 4.2 · p. 25
  • For sufficiently small ν, Lax shocks oscillate on the left and are monotone on the right when μ > 0, with the orientation reversed when μ < 0.The oscillation boundary is determined by the transition between real and complex eigenvalues, and the predicted distinction is observed numerically.Source: §4.2 · p. 26

5. Riemann problem for the mKdV equation.

This section classifies mKdV Riemann-problem solutions and their wave patterns, including DSWs, rarefactions, kinks, compound waves, and contact DSW structure. It develops traveling-wave and modulation building blocks and compares purely dispersive waves with diffusive-dispersive shocks.

5.1.1. Traveling waves.

  • For μ > 0, mKdV traveling waves include kinks connecting equilibria; for both signs of μ, periodic waves have solitons of both polarities, while μ < 0 also admits a non-KdV trigonometric family.The kink speed agrees with the classical shock speed.Source: §5.1.1 · p. 29

5.1.2. Modulation equations.

  • The mKdV-Whitham system is non-strictly hyperbolic and not genuinely nonlinear, and its non-strict hyperbolicity underlies contact DSWs absent from the scalar mKdVB shock theory.The modulation mapping to KdV variables can be non-unique, with the selected branch set by initial or boundary conditions.Source: §5.1.2 · pp. 31–32

5.1.3. Classical mKdV DSW solutions and admissibility conditions.

  • Classical mKdV DSWs require causality and convexity admissibility, with the initial-state interval avoiding u = 0; when causality alone fails, a rarefaction wave forms instead.Unlike the Lax shock, the DSW's zero-dispersion weak limit is its period-mean value.Source: §5.1.3 · pp. 32, 34

5.1.4. Non-classical DSWs.

  • For μ > 0, kinks are undercompressive DSWs; for μ < 0, contact DSWs connect conjugate states through oscillations and are dispersive counterparts of single-sided contact discontinuities.The contact DSW's special modulation has zero elliptic parameter but nonzero wave amplitude.Source: §5.1.4 · pp. 34–35
  • A CDSW+ has a nonlinear trigonometric carrier, connects only conjugate states u+ = −u− with u− > 0, and has a bright algebraic soliton at its leading edge.Its modulation solution is a double-characteristic fan; nonstrict hyperbolicity permits this structure, and the triple-characteristic leading edge can match to a rarefaction.Source: §5.1.4 · pp. 35–36
  • Near the harmonic edge, the CDSW envelope scales as √(x − x−), unlike the linear scaling of the wave amplitude; simulations may obscure this distinction.The cited passage attributes the difference to contributions absent from the Whitham multiple-scale solution that may matter near the harmonic edge.Source: §5.1.4 · p. 36

5.2. Riemann problem classification.

  • The mKdV initial-state plane is divided into eight wave-pattern sectors by u− = 0, u+ = 0, u+ = u−, and u+ = −u−, with boundaries symmetric under a π/4 rotation.Source: §5.2 · p. 36

5.2.1. Classification for µ > 0.

  • For µ > 0, failed DSW causality with convexity satisfied gives rarefactions, while the line u+ = −u− yields a kink or antikink; crossing u+ = 0 peels a kink from DSW−.As u+ decreases through zero, the remaining wave train becomes DSW+.Source: §5.2.1 · pp. 36–37
  • For µ > 0, −u− < u+ < 0 gives a widening kink|DSW+ pair, whereas u+ < −u− < 0 gives a kink followed by a rarefaction across a widening constant-state interval.Source: §5.2.1 · pp. 36–37

5.2.2. Classification for µ < 0.

  • For µ < 0, −u− < u+ < 0 produces a compound CDSW+DSW+; for u+ < −u− < 0, a fully developed CDSW+ connects at its leading edge to a rarefaction.The compound wave has a partial CDSW+ attached to a DSW+; their separating characteristic is s* = 6u−^2.Source: §5.2.2 · p. 37

6. Comparison.

The comparison traces how nonconvexity changes the relation between diffusive-dispersive traveling waves and purely dispersive waves. The zero-diffusion correspondence is multivalued, and small but nonzero diffusion produces a transient evolution before the diffusive-dispersive balance emerges.

6.1. Shock structure.

  • Diffusion damps dispersive oscillations, whereas purely dispersive dynamics continually generate large-amplitude oscillations; undercompressive shocks and kinks remain monotone.For nonconvex flux, polarity can also change without changing the dispersion sign.Source: §6.1 · p. 38

6.2. Characteristic diagrams.

  • A contact DSW-rarefaction has a leading triple characteristic and maintains Γ2 = Γ3 across the contact DSW.In a contact DSW-DSW complex, the double characteristics continue to the interface with the classical DSW.Source: §6.2 · pp. 40–41
  • For positive dispersion, undercompressive shocks map to kinks, which are non-modulated traveling waves; kink characteristics pass through without deflection.The K|DSW± solution combines a kink and a modulated DSW.Source: §6.2 · pp. 41–42, §6.3 · p. 44

6.3. Zero Diffusion Limit.

  • In same-sign state regions, diffusive-dispersive Lax shocks map to DSWs as diffusion vanishes, while rarefactions have the same leading-order regularization.For positive dispersion, the mapping also includes Σ|S± → K|DSW± and Σ|R → K|R.Source: §6.3 · pp. 43–44
  • For negative dispersion, shock-rarefactions can map to contact-DSW-rarefactions, while Lax shocks can map to contact DSWs, hybrid contact-DSW-DSWs, or pure DSWs.These mappings are multivalued, and the distinct diffusive and nondiffusive intermediate states reflect different shock and contact-DSW conditions.Source: §6.3 · p. 45

6.4. Time scales and critical scalings.

  • Diffusion and dispersion balance on the time scale t ∼ |μ|/ν, following a transient interval in which dispersion dominates.As ν approaches zero with fixed μ, diffusive effects take longer to establish themselves.Source: §6.4 · p. 46
  • With μ = −1 and ν = 0.01, initial dynamics resemble the purely dispersive case, but by t = 100 the diffusive-dispersive behavior has emerged.Between the initial DSW formation and that long-time state, a backward-propagating wave packet leaves a steady shock adjacent to a rarefaction.Source: §6.4 · p. 47

6.5. Small-diffusion regime and modulation equations.

  • Small-diffusion perturbed modulation theory can describe classical oscillating mKdVB shocks, but oscillatory contact shocks in negative-dispersion shock-rarefaction complexes require further analysis.The proposed mKdVB/KdVB modulation-system relation follows the same surjective mapping as their purely dispersive counterparts.Source: §6.5 · pp. 47–48

7. Conclusions.

The conclusions compare zero-diffusion limits for nonconvex dispersive and diffusive-dispersive shock solutions. They identify contact DSWs and kinks as dispersive counterparts of features in the mixed-regularization theory, while noting limits to the present analysis.

7. Conclusions.

  • For nonconvex flux, the zero-diffusion mapping is multivalued, unlike the bijective mapping for convex-flux KdVB and KdV.For negative dispersion, Lax shocks may map to contact DSWs or DSWs, and the mapping has some ambiguity.Source: §7 · p. 48
  • A contact DSW is a purely dispersive feature linked to nonstrict hyperbolicity of the Whitham modulation equations; its characteristics can pass through or become tangent to an adjacent wave.This contrasts with negative-dispersion diffusive-dispersive shocks, which satisfy Lax entropy conditions.Source: §7 · p. 48
  • The zero-diffusion limit includes a transient window in which dispersive-dominated dynamics transition to a diffusive-dispersive balance.The paper identifies relevant time scales to interpret this transition.Source: §7 · p. 49
  • For positive dispersion, an undercompressive shock maps to a monotone kink, whose characteristics pass through without deflection.Unlike classical DSWs, kinks are traveling waves without expanding, oscillatory profiles.Source: §7 · p. 49
  • The analysis focuses on scalar models; extending the theory to fully nonlinear, bidirectional nonconvex Eulerian systems remains a major development.The counterpart zero-diffusion theory for nonconvex p-systems is not presently available.Source: §7 · p. 49

Appendix A. Traveling wave solutions of the mKdV equation.

The appendix classifies mKdV traveling-wave solutions by dispersion sign and limiting root configurations. It highlights features associated with the nonconvex flux, including both-polarity solitons and nonlinear trigonometric waves.

Appendix A. Traveling wave solutions of the mKdV equation.

  • For positive dispersion, mKdV admits solitons of both polarities, unlike KdV, where admissible solitary-wave polarity is tied to dispersion sign.The counterpart dark solutions follow from the reflection transformation.Source: Appendix A · pp. 50, 52
  • When the potential has two double roots, the traveling-wave solution is a kink or antikink with propagation speed U = u1^2.The kink connects u4 = −u1 to u1, and its speed agrees with the classical shock speed.Source: Appendix A · p. 51
  • The appendix also gives bright and dark soliton limits for negative dispersion, with elevation or depression relative to their constant backgrounds.The bright soliton amplitude is a = u4 − u2; the dark soliton arises when u3 → u4.Source: Appendix A · pp. 50–51
  • For negative dispersion, the nonlinear trigonometric solution has no KdV analogue and can have order-unity amplitude.Its existence is attributed to the quartic potential associated with the nonconvex hyperbolic flux.Source: Appendix A · p. 52
  • Periodic solutions associated with complex-conjugate roots are modulationally unstable and do not arise in the Riemann problem.The appendix therefore does not describe them further.Source: Appendix A · p. 52

Appendix B. Numerical method.

The appendix describes a Fourier pseudospectral integrating-factor method with RK4 time stepping for the KdV(B) and mKdV(B) equations. It reports validation against exact traveling waves and numerical boundary and conservation checks.

Appendix B. Numerical method.

  • The computations use a high-order integrating-factor pseudospectral Fourier method with fourth-order Runge-Kutta time stepping.The method is motivated by Trefethen’s approach to KdV and uses differing constant boundary values.Source: Appendix B · pp. 52–53
  • The third-derivative stiffness is absorbed into a transformation to stabilize time stepping with larger Δt; the second-derivative term is excluded from that transformation to avoid numerical instability.The transformed Fourier coefficients are advanced with RK4.Source: Appendix B · p. 53
  • The reconstruction is spectrally accurate while the transformed solution remains localized within [−L, L].It uses FFT-computed Fourier coefficients and trapezoidal-rule integration.Source: Appendix B · p. 53
  • The method was tested against exact traveling waves, with the conserved quantity maintained within 10−13 of its nominal value across simulations.Boundary values remained within 10−4 of u±.Source: Appendix B · pp. 53–54
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