Shrira Equation: Dual Wave Models

• Shrira Equation is a dual-natured concept in fluid dynamics, denoting both an exact deep-water dispersion relation for gravity waves and a two-dimensional nonlocal dispersive PDE.
• The deep-water formulation uses the Rayleigh equation and Green’s function methods to account for arbitrary vertical shear and wave sampling throughout the current profile.
• The nonlocal PDE model employs the Hilbert transform to introduce anisotropic dispersion, underpinning analyses of solitary waves, well-posedness, and instability.

The name Shrira equation is used in two distinct senses in the literature. In linear wave–current interaction, it denotes the exact deep-water dispersion relation for linear surface gravity waves propagating on a horizontally uniform current with arbitrary vertical shear, a result attributed to Shrira (1993) and recovered as the deep-water limit of an exact finite-depth formulation in "Exact dispersion relation for linear surface waves on arbitrary vertical shear" (Heinrich et al., 27 Apr 2026). In dispersive partial differential equations, the same name denotes a two-dimensional anisotropic nonlocal evolution equation, commonly written

$$\partial_t u-\mathcal{H}_x\partial_x^2u-\mathcal{H}_x\partial_y^2u+u\partial_x u=0,$$

introduced as a simplified model for weakly nonlinear long-wave perturbations on the background of a boundary-layer type plane-parallel shear flow (Riaño, 2020). A generalized solitary-wave version,

$u_t-\mathscr H\Delta u+(f(u))_x=0,$

is standard in the nonlinear theory (Esfahani et al., 2017). The terminological overlap is substantive rather than merely notational: the first object is an exact linear dispersion relation, whereas the second is a nonlinear two-dimensional dispersive model.

1. Dual usage and scope of the term

The linear-water-wave usage and the dispersive-PDE usage refer to mathematically different objects. In the first, the Shrira equation is a scalar implicit relation between frequency, wavenumber, and a prescribed shear profile. In the second, it is an evolution equation for a scalar field $u(x,y,t)$ with nonlocal anisotropic dispersion through the Hilbert transform in the longitudinal direction. The overlap reflects a common physical background in shear flows, but the analytical frameworks are different: Rayleigh theory and boundary-value propagation in the former, nonlocal dispersive PDE methods in the latter (Heinrich et al., 27 Apr 2026).

This distinction matters for interpretation. The deep-water Shrira equation is not an asymptotic approximation within deep water; it is the exact deep-water member of a more general exact finite-depth family. Conversely, the two-dimensional Shrira PDE is not a dispersion relation but an effective nonlinear model. A further source of confusion is the appearance of Shrira-associated reduced models for trapped waves on jet currents. The single-mode modified nonlinear Schrödinger equation derived in "Nonlinear dynamics of trapped waves on jet currents and rogue waves" (Shrira et al., 2014) is explicitly described as a weakly nonlinear reduction within the Shrira–Slunyaev trapped-wave modal framework, rather than as the Shrira equation itself.

2. The deep-water exact dispersion relation on arbitrary vertical shear

In the linear water-wave setting, one considers an inviscid incompressible fluid of constant density with undisturbed free surface at $z=0$, flat bed at $z=-h$, and a horizontally uniform background current $\mathbf U(z)$. Infinitesimal-amplitude surface gravity waves with horizontal wavevector $\mathbf k$, frequency $\omega$, and phase factor $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$ are superposed on that current. The intrinsic frequency is

$$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$

After linearization and elimination of pressure and horizontal velocities, the vertical-velocity amplitude $u_t-\mathscr H\Delta u+(f(u))_x=0,$0 satisfies the inviscid Rayleigh equation

$u_t-\mathscr H\Delta u+(f(u))_x=0,$1

with the free-surface condition

$u_t-\mathscr H\Delta u+(f(u))_x=0,$2

The structure of these equations separates surface shear, which enters through $u_t-\mathscr H\Delta u+(f(u))_x=0,$3, from bulk curvature, which enters through $u_t-\mathscr H\Delta u+(f(u))_x=0,$4 (Heinrich et al., 27 Apr 2026).

In deep water, $u_t-\mathscr H\Delta u+(f(u))_x=0,$5 and the bottom condition is replaced by decay as $u_t-\mathscr H\Delta u+(f(u))_x=0,$6. The exact deep-water relation identified with Shrira’s result is

$u_t-\mathscr H\Delta u+(f(u))_x=0,$7

with

$u_t-\mathscr H\Delta u+(f(u))_x=0,$8

An equivalent form is

$u_t-\mathscr H\Delta u+(f(u))_x=0,$9

This formulation makes explicit that arbitrary shear cannot, in general, be reduced to a single Doppler shift by an effective current. For $u(x,y,t)$0, one recovers the Doppler-shifted still-water law; for constant-vorticity profiles, $u(x,y,t)$1 and all shear effects enter through the surface boundary condition alone; for genuinely curved profiles, the wave samples the whole current profile through the depth integral (Heinrich et al., 27 Apr 2026).

3. Finite-depth generalization and exact propagator structure

The finite-depth generalization proceeds by rewriting the Rayleigh problem in a Green’s-function framework. With $u(x,y,t)$2 solving

$u(x,y,t)$3

together with the bed condition, the Rayleigh equation becomes a Lippmann–Schwinger-type integral equation,

$u(x,y,t)$4

This makes the curvature term appear only through the effective potential

$u(x,y,t)$5

The exact implicit dispersion relation can then be written as

$u(x,y,t)$6

or equivalently in an explicit integral form containing only $u(x,y,t)$7, $u(x,y,t)$8, $u(x,y,t)$9, and $z=0$0 (Heinrich et al., 27 Apr 2026).

A second exact representation recasts the Rayleigh equation as a first-order $z=0$1 system and expresses the depth propagator as a path-ordered exponential,

$z=0$2

Its significance is that the full dependence on $z=0$3 is isolated in a single noncommuting propagation operator. This creates a natural starting point for systematic approximations by Dyson or Magnus series. In the deep-water limit, the finite-depth kernel reduces to

$z=0$4

the bare logarithmic derivative $z=0$5 tends to $z=0$6, bottom-reflection terms vanish exponentially, and the exact finite-depth relation reduces precisely to the deep-water Shrira equation. This reduction is one of the central structural results of (Heinrich et al., 27 Apr 2026).

4. The two-dimensional nonlocal dispersive PDE

In the PDE literature, the Shrira equation is the Cauchy problem

$z=0$7

posed on either $z=0$8 or $z=0$9. The Hilbert transform in the $z=-h$0-direction is defined on $z=-h$1 by

$z=-h$2

and analogously in the periodic setting. The associated $z=-h$3-fractional derivative is

$z=-h$4

Its linear symbol is

$z=-h$5

The equation is scale-invariant under

$z=-h$6

and is therefore $z=-h$7-critical (Riaño, 2020).

A generalized form studied in the solitary-wave theory is

$z=-h$8

with $z=-h$9 and $\mathbf U(z)$0 the Hilbert transform in $\mathbf U(z)$1. The classical quadratic case is recovered by $\mathbf U(z)$2. This model is described as a two-dimensional anisotropic nonlocal analogue of the Benjamin–Ono equation

$\mathbf U(z)$3

For real solutions, the conserved mass is

$\mathbf U(z)$4

and one natural energy is

$\mathbf U(z)$5

In the instability literature, the same equation is also written as

$\mathbf U(z)$6

with $\mathbf U(z)$7, emphasizing the connection to the fractional operator $\mathbf U(z)$8 (Esfahani et al., 2017, Méndez et al., 18 Sep 2025).

5. Well-posedness, weighted theory, and solitary waves

Local well-posedness for the Shrira PDE is established in Sobolev spaces $\mathbf U(z)$9, $\mathbf k$0, for $\mathbf k$1, and also in the anisotropic energy-adapted space $\mathbf k$2 with norm

$\mathbf k$3

The same framework yields results in weighted anisotropic Sobolev spaces

$\mathbf k$4

their mean-zero variants $\mathbf k$5, and the mixed space $\mathbf k$6. The paper further derives unique-continuation consequences, including the identity

$\mathbf k$7

and the corollary that if $\mathbf k$8 for a.e. $\mathbf k$9, then $\omega$0 (Riaño, 2020).

For the generalized equation, solitary waves are sought in the form

$\omega$1

which yields

$\omega$2

or, after integrating in $\omega$3,

$\omega$4

The natural energy space is

$\omega$5

with norm

$\omega$6

Under assumptions (A1)–(A4), the stationary problem has a nontrivial solution $\omega$7, obtained by a mountain-pass argument. Under (A1)–(A5), ground states exist as minimizers on the Nehari manifold. The paper also proves that any solitary-wave solution belongs to $\omega$8 for $\omega$9; if $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$0, $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$1, then $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$2; and for $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$3, one has $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$4. Under analytic-type assumptions on $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$5, the solitary wave is real analytic. The decay is anisotropic: $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$6 which yields the qualitative rates $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$7 in $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$8 and $e^{i(\mathbf k\cdot \mathbf x-\omega t)}$9 in $$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$0 (Esfahani et al., 2017).

6. Critical traveling waves, instability, and related descendants

The modern instability theory treats the Shrira equation as the two-dimensional $$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$1 case of the fractional Zakharov–Kuznetsov family

$$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$2

Traveling waves have the form

$$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$3

with profile equation

$$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$4

and scaling

$$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$5

The ground state $$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$6 is cited as a unique positive radial ground state, up to translation, with $$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$7 and polynomial decay $$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$8. Under a conditional $$\sigma(z)=\omega-\mathbf k\cdot \mathbf U(z), \qquad \sigma_0=\omega-\mathbf k\cdot \mathbf U(0).$$9 well-posedness assumption, "A monotonicity formula for the fractional Laplacian and instability results for the Shrira equation" proves conditional orbital instability of $u_t-\mathscr H\Delta u+(f(u))_x=0,$00 in the $u_t-\mathscr H\Delta u+(f(u))_x=0,$01-critical regime. The key analytical input is the monotonicity inequality

$u_t-\mathscr H\Delta u+(f(u))_x=0,$02

which yields directional weighted $u_t-\mathscr H\Delta u+(f(u))_x=0,$03 control and replaces the pointwise tail estimates used in previous instability analyses (Méndez et al., 18 Sep 2025).

A separate, but conceptually adjacent, development is the trapped-wave modal theory on jet currents associated with Shrira and Slunyaev. For a wave field narrowband in frequency but not necessarily with narrow angular distributions, (Shrira et al., 2014) derives the one-dimensional modified nonlinear Schrödinger equation

$u_t-\mathscr H\Delta u+(f(u))_x=0,$04

The paper states explicitly that this is a reduction and application of the Shrira–Slunyaev trapped-mode framework, not an independent equation called the Shrira equation. Its presence in the literature nonetheless illustrates a broader pattern: Shrira’s name is attached not only to the exact deep-water dispersion relation and the two-dimensional nonlocal PDE, but also to a family of modal and envelope reductions for wave–current interaction (Shrira et al., 2014).