Stable Shadow Wave Solution in PDEs

• Stable shadow wave solutions are singular constructs in nonlinear PDEs that use thin, diverging layers to represent concentrated masses or unbounded cores.
• They are built from piecewise classical or constant profiles outside narrow regions and matched with diverging intermediate states under entropy or overcompressibility criteria.
• Applications include discontinuous-flux conservation laws, radially symmetric pressureless gas dynamics, and Euler–Poisson collapse, where stability is maintained via weak residual control and entropy inequalities.

Searching arXiv for the cited papers and closely related work on shadow waves and stability. A stable shadow wave solution is a singular solution concept for nonlinear PDEs in which a concentrating structure is represented by a family of approximate solutions with a thin layer of width $O(\varepsilon)$ and amplitude that diverges as $\varepsilon\to 0$, so that the limit carries a Dirac mass or an unbounded core. In the literature represented here, the term appears most concretely in conservation laws, pressureless gas dynamics, and spherically symmetric gravitational collapse. Across these settings, a shadow wave is not a bounded classical weak solution; it is an admissible singular limit selected by distributional balance, overcompressibility or entropy, nonnegativity of concentrated mass, or by the requirement that the weak residuals remain $O(\varepsilon)$ as $\varepsilon\to 0$ (Krunić et al., 2021, Nedeljkov et al., 2016, Nedeljkov et al., 17 Aug 2025).

1. Conceptual definition and structural features

The common construction is a piecewise classical or piecewise constant profile outside a narrow region, together with an intermediate state inside that region whose magnitude diverges as the layer thickness shrinks. The limit is designed to retain a finite concentrated mass. In one class of interface problems, the limit is a delta concentration at $x=0$; in radial pressureless gas dynamics, it is a Dirac mass on a moving sphere; in Euler–Poisson collapse, it is an unbounded core at the origin. This suggests that “stable” does not denote a single universal property, but rather a model-dependent admissibility and persistence criterion.

Setting	Shadow-wave object	Stability or selection principle
Discontinuous-flux scalar conservation law	Stationary interface concentration $k(t)\delta(x)$	Overcompressive admissibility and uniqueness of limit strength
Radial pressureless gas dynamics	Concentrated mass on $r=R+c(t)$	Weak balance laws, $\sigma(t)\ge 0$, entropy inequality
1D pressureless gas tracking	Delta mass on a moving shock curve	Overcompressibility, interaction rules, entropy decrease
Euler–Poisson collapse	Core $r<\varepsilon t$ with $\rho_\varepsilon\sim \varepsilon^{-3}$	Weak residuals $\varepsilon\to 0$0, $\varepsilon\to 0$1

A central distinction is between the approximate profile and its limit. In the scalar two-flux problem, the approximate solution is a net of piecewise constant functions concentrated near the interface, while the limit has the form of a jump plus a delta mass. In radial pressureless gas, the approximate layer has density $\varepsilon\to 0$2, and its limit is a moving spherical concentration. In Euler–Poisson collapse, finite mass in a shrinking core of volume $\varepsilon\to 0$3 forces the density to scale like $\varepsilon\to 0$4, producing an unbounded origin singularity rather than a bounded delta-shock profile (Krunić et al., 2021, Nedeljkov et al., 2016, Nedeljkov et al., 17 Aug 2025).

2. Interface shadow waves for discontinuous-flux conservation laws

For the scalar conservation law

$\varepsilon\to 0$5

with

$\varepsilon\to 0$6

the classical interface condition requires

$\varepsilon\to 0$7

The central case studied is one in which this equality cannot be achieved while maintaining the required admissibility. The paper focuses on geometries such as $\varepsilon\to 0$8 on the whole domain, with one flux of convex type and the other of concave type, producing a Rankine–Hugoniot deficit

$\varepsilon\to 0$9

In that regime, a bounded weak interface shock is unavailable, and the deficit is absorbed by a singular concentration at the origin (Krunić et al., 2021).

The shadow wave is introduced as a family

$O(\varepsilon)$0

with intermediate states that may blow up as $O(\varepsilon)$1. Its strength is

$O(\varepsilon)$2

If $O(\varepsilon)$3, then the limit has the form

$O(\varepsilon)$4

The overcompressive condition

$O(\varepsilon)$5

is used as the admissibility criterion. Under the stated growth hypotheses on $O(\varepsilon)$6 and $O(\varepsilon)$7, the limit is

$O(\varepsilon)$8

The paper distinguishes two singular limits. A delta shock wave occurs when the singular limit is a Dirac delta with the minor components remaining finite. A singular shock wave occurs when one intermediate state blows up so strongly that $O(\varepsilon)$9 and $\varepsilon\to 0$0. The same Rankine–Hugoniot deficit can therefore be resolved either by a delta shock or by a stronger singular shock, depending on the asymptotic growth of the fluxes. The stability statement is correspondingly limited and precise: in the overcompressive setting, the stationary shadow wave is unique in the weak/distributional sense at the level of its limit strength $\varepsilon\to 0$1, but the detailed asymptotic structure of the intermediate states is not unique in all cases (Krunić et al., 2021).

3. Radially symmetric shadow waves in pressureless gas dynamics

For the multidimensional pressureless Euler system

$\varepsilon\to 0$2

the radially symmetric reduction is

$\varepsilon\to 0$3

A radially symmetric shadow wave is introduced as a thin $\varepsilon\to 0$4-layer around a moving spherical interface $\varepsilon\to 0$5, with density

$\varepsilon\to 0$6

In the limit, the density becomes

$\varepsilon\to 0$7

so the singular mass is located on a sphere rather than at a point (Nedeljkov et al., 2016).

The weak formulation yields exact evolution laws for the concentrated mass. With

$\varepsilon\to 0$8

and

$\varepsilon\to 0$9

the shadow-wave equations are

$x=0$0

$x=0$1

A key structural consequence is that if $x=0$2 on an interval, then necessarily

$x=0$3

so the intermediate layer moves with the wave speed.

Stability is encoded by three simultaneous requirements. First, the profile must satisfy the weak radial balance laws. Second, the concentrated mass must remain nonnegative: $x=0$4 Third, it must satisfy the entropy inequality derived from kinetic energy. The entropy condition for a shadow wave is

$x=0$5

equivalently,

$x=0$6

This is the paper’s criterion for selecting dissipative solutions.

For constant speed $x=0$7, the wave speed satisfies a quadratic relation with two roots, but only

$x=0$8

is both physical and dissipative. In the pseudo-Riemann case with $x=0$9 and $k(t)\delta(x)$0, the resulting spherical delta shock has

$k(t)\delta(x)$1

The paper concludes that the entropy condition selects the unique admissible shock speed, while nonentropy physical shadow waves with nonconstant speed can exist but are not stable in the entropy sense (Nedeljkov et al., 2016).

4. Shadow-wave tracking for one-dimensional pressureless gas

In the one-dimensional pressureless gas system

$k(t)\delta(x)$2

shadow waves are used as singular approximations of delta-shock type solutions. The construction adapts Wave Front Tracking to a setting in which the solution may contain Dirac deltas supported on moving curves. A shadow wave is defined by a central shadow wave line $k(t)\delta(x)$3, thin left and right layers of size $k(t)\delta(x)$4, and a finite strength obtained by multiplying layer width and diverging density. A shadow wave is called simple if its speed and intermediate states are constant (Nedeljkov et al., 2019).

The Riemann solver has three cases. If $k(t)\delta(x)$5, the solution is a contact discontinuity. If $k(t)\delta(x)$6, the solution contains vacuum. If $k(t)\delta(x)$7, the solution is a simple shadow wave. For the singular case, the shadow wave speed $k(t)\delta(x)$8 and strength $k(t)\delta(x)$9 are governed by

$r=R+c(t)$0

The front location is

$r=R+c(t)$1

A major extension is the treatment of initial data containing Dirac delta functions at interaction points. When two incoming singular waves interact, the new initial strength is the sum of the incoming strengths,

$r=R+c(t)$2

and the new speed is the momentum-weighted average

$r=R+c(t)$3

This provides a closed interaction rule for delta masses. The admissibility criterion is overcompressibility: $r=R+c(t)$4 The paper states that overcompressibility is preserved under interactions.

The resulting approximate solution is global under the stated monotonicity hypotheses, and a subsequence converges weakly to a signed Radon measure: $r=R+c(t)$5 The entropy pair

$r=R+c(t)$6

yields a nonpositive entropy production across a shadow wave,

$r=R+c(t)$7

and the total entropy decreases after interaction between two shadow waves. In this setting, stability is therefore algorithmic and admissibility-based: the singular waves remain overcompressive, their interactions are closed under the update rules, and the approximations admit a measure-valued limit (Nedeljkov et al., 2019).

5. Stable shadow waves in spherically symmetric gravitational collapse

A distinct notion of stable shadow wave appears in the three-dimensional Euler–Poisson collapse model with density-dependent viscosity. The starting system is

$r=R+c(t)$8

with

$r=R+c(t)$9

Under spherical symmetry, the radial equations become

$\sigma(t)\ge 0$0

$\sigma(t)\ge 0$1

The solution is modeled by a core-plus-exterior ansatz,

$\sigma(t)\ge 0$2

Here the core $\sigma(t)\ge 0$3 is the shadow-wave region (Nedeljkov et al., 17 Aug 2025).

The decisive scaling comes from mass balance in the shrinking core. Since

$\sigma(t)\ge 0$4

finite nonzero core mass requires

$\sigma(t)\ge 0$5

The core velocity is taken as

$\sigma(t)\ge 0$6

Theorem 1 states that the profile is an approximate spherically symmetric solution of the full three-dimensional system as $\sigma(t)\ge 0$7 if

$\sigma(t)\ge 0$8

$\sigma(t)\ge 0$9

and if the exterior satisfies

$r<\varepsilon t$0

Without viscosity, the same core scaling remains valid and the outer requirement weakens to

$r<\varepsilon t$1

The notion of stability is explicit and asymptotic. For every test function $r<\varepsilon t$2, the weak residuals satisfy

$r<\varepsilon t$3

Thus the shadow wave is stable in the sense that the core-plus-exterior profile is a consistent approximate weak solution, with all dangerous contributions controlled as $r<\varepsilon t$4. The paper then studies a self-similar outer ansatz

$r<\varepsilon t$5

defines

$r<\varepsilon t$6

and proves the global inequalities

$r<\varepsilon t$7

These properties keep the exterior in the collapsing regime. In this model, therefore, a stable shadow wave solution is a shrinking singular core with unbounded density at the origin, matched to a controlled outer radial flow (Nedeljkov et al., 17 Aug 2025).

6. Terminological scope and non-equivalent usages

The phrase “stable shadow wave solution” is not uniform across the literature, and several nearby uses of “shadow,” “wave,” and “stability” are mathematically distinct. In the fourth-order dispersive nonlinear Schrödinger equation, the proved result is orbital stability of the standing wave

$r<\varepsilon t$8

in $r<\varepsilon t$9. The paper explicitly states that this is a genuine standing wave and its orbital stability, and that “shadow wave” is at most an informal description of remaining close to the symmetry orbit; it is not a separate singular solution concept (Natali et al., 2015).

A related distinction appears in the three-dimensional quadratic Zakharov–Kuznetsov equation, where the result is asymptotic stability of the solitary wave $\rho_\varepsilon\sim \varepsilon^{-3}$0. There the solution converges, after modulation, to a rescaling and shift of the ground state in a rightward-shifting conic region. This is stronger than orbital stability, but it is still not a shadow-wave construction in the sense of a thin singular layer approximating a delta concentration (Farah et al., 2020).

The terminology is also unrelated to gravitational and optical uses of “shadow.” In four-dimensional Einstein–Gauss–Bonnet gravity, the shadow is determined by unstable null circular geodesics on the photon sphere, with

$\rho_\varepsilon\sim \varepsilon^{-3}$1

and the discussion concerns black-hole size bounds rather than singular weak solutions (Guo et al., 2020). In compact-object imaging, stable Proca stars can mimic a Schwarzschild shadow in an astrophysical environment through disk truncation at an effective radius $\rho_\varepsilon\sim \varepsilon^{-3}$2, despite the absence of a light ring; again, this concerns optical appearance, not shadow waves in PDE theory (Herdeiro et al., 2021).

A final source of ambiguity is computational imaging. “Detail-Preserving Latent Diffusion for Stable Shadow Removal” uses a pre-trained Stable Diffusion model for shadow removal in images, where “stable” refers to robust latent-space inference and “shadow” refers to illumination artifacts. This usage is entirely separate from the singular-solution notion discussed above (Xu et al., 2024).

Within PDE theory proper, the most precise meaning of a stable shadow wave solution is therefore the one developed for discontinuous-flux conservation laws, pressureless gas dynamics, and Euler–Poisson collapse: a singular approximate profile whose limit captures concentrated mass or an unbounded core, and whose admissibility is fixed by weak balance, entropy, overcompressibility, or residual control (Krunić et al., 2021, Nedeljkov et al., 2016, Nedeljkov et al., 17 Aug 2025).