--- title: 'Shrira Equation: Dual Wave Models' url: https://www.emergentmind.com/topics/shrira-equation type: topic --- # Shrira Equation: Dual Wave Models 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" [2604.24484]. 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 [2005.09184]. A generalized solitary-wave version, \[ u_t-\mathscr H\Delta u+(f(u))_x=0, \] is standard in the nonlinear theory [1705.01627]. 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 [2604.24484]. 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" [1401.4455] 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 \(w(z)\) satisfies the inviscid Rayleigh equation \[ w''-k^2 w=\frac{\mathbf k\cdot \mathbf U''(z)}{\sigma(z)}\,w, \] with the free-surface condition \[ \bigl[\sigma_0^2\,w'(0)-\bigl(gk^2+\sigma_0\,\mathbf k\cdot \mathbf U'(0)\bigr)w(0)\bigr]=0. \] The structure of these equations separates surface shear, which enters through \(\mathbf U'(0)\), from bulk curvature, which enters through \(\mathbf U''(z)\) [2604.24484]. In deep water, \(h\to\infty\) and the bottom condition is replaced by decay as \(z\to-\infty\). The exact deep-water relation identified with Shrira’s result is \[ \bigl[1+I_g(\omega,\mathbf k)\bigr]\sigma_0^2 +\sigma_0\,\mathbf k\cdot \mathbf U'(0)-gk=0, \] with \[ I_g(\omega,\mathbf k) = \int_{-\infty}^{0} \frac{\mathbf k\cdot \mathbf U''(z)}{k\,\sigma(z)} \frac{\Phi(z)}{\Phi(0)} \,e^{kz}\,dz. \] An equivalent form is \[ \sigma_0^2 +\sigma_0\,\frac{\mathbf k\cdot \mathbf U'(0)}{k} -g = -\sigma_0^2 \int_{-\infty}^{0} \frac{\mathbf k\cdot \mathbf U''(z)}{k\,\sigma(z)} \,\frac{w(z)}{w(0)}\,e^{kz}\,dz. \] This formulation makes explicit that arbitrary shear cannot, in general, be reduced to a single Doppler shift by an effective current. For \(\mathbf U(z)=\mathbf U_0\), one recovers the Doppler-shifted still-water law; for constant-vorticity profiles, \(\mathbf U''=0\) 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 [2604.24484]. ## 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 \(G(z,\zeta)\) solving \[ (\partial_z^2-k^2)G(z,\zeta)=\delta(z-\zeta), \] together with the bed condition, the Rayleigh equation becomes a Lippmann–Schwinger-type integral equation, \[ w(z)=w^{(0)}(z)+\int_{-h}^0 G(z,\zeta)\,\frac{\mathbf k\cdot \mathbf U''(\zeta)}{\sigma(\zeta)}\,w(\zeta)\,d\zeta. \] This makes the curvature term appear only through the effective potential \[ V(z)=\frac{\mathbf k\cdot \mathbf U''(z)}{\sigma(z)}. \] The exact implicit dispersion relation can then be written as \[ \Delta(\mathbf k,\omega)= \sigma_0^2\,\Phi'(0)-\Bigl(gk^2+\sigma_0\,\mathbf k\cdot \mathbf U'(0)\Bigr)\Phi(0)=0, \] or equivalently in an explicit integral form containing only \(\mathbf U(z)\), \(k\), \(h\), and \(\omega\) [2604.24484]. A second exact representation recasts the Rayleigh equation as a first-order \(2\times2\) system and expresses the depth propagator as a path-ordered exponential, \[ \mathcal P\exp\!\left[\int_{z_1}^{z_2}\! \mathbf M(s)\,ds\right]. \] Its significance is that the full dependence on \(\mathbf U''(z)\) 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 \[ G_\infty(z,\zeta)=\frac{1}{2k}\Bigl(e^{-k|z-\zeta|}-e^{k(z+\zeta)}\Bigr), \] the bare logarithmic derivative \(k\coth(kh)\) tends to \(k\), 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 [2604.24484]. ## 4. The two-dimensional nonlocal dispersive PDE In the PDE literature, the Shrira equation is the Cauchy problem \[ \begin{cases} \partial_t u -\mathcal{H}_x\partial_x^2u-\mathcal{H}_x\partial_y^2u+u\partial_{x} u=0,\\ u(x,0)=u_0, \end{cases} \] posed on either \(\mathbb R^2\) or \(\mathbb T^2\). The Hilbert transform in the \(x\)-direction is defined on \(\mathbb R^2\) by \[ \mathcal{F}(\mathcal{H}_x \phi )(\xi,\eta)=-i\sign(\xi)\widehat{\phi}(\xi,\eta), \] and analogously in the periodic setting. The associated \(x\)-fractional derivative is \[ \mathcal{F}(D_x^{\pm 1/2}u)(\xi,\eta)=|\xi|^{\pm 1/2}\widehat{u}(\xi,\eta). \] Its linear symbol is \[ \widetilde{\omega}(\xi,\eta)=\sign(\xi)\xi^2+\sign(\xi)\eta^2. \] The equation is scale-invariant under \[ u_{\lambda}(x,y,t)=\lambda u(\lambda x,\lambda y, \lambda^2 t), \] and is therefore \(L^2\)-critical [2005.09184]. A generalized form studied in the solitary-wave theory is \[ u_t-\mathscr H\Delta u + (f(u))_x=0, \] with \(\Delta=\partial_x^2+\partial_y^2\) and \(\mathscr H\) the Hilbert transform in \(x\). The classical quadratic case is recovered by \(f(u)=u^2\). This model is described as a two-dimensional anisotropic nonlocal analogue of the Benjamin–Ono equation \[ u_t-\mathscr H u_{xx}+2u u_x=0. \] For real solutions, the conserved mass is \[ M(u)=\int u^2(x,y,t)\, dxdy, \] and one natural energy is \[ \widetilde{E}(u)=\frac{1}{2} \int |D_x^{1/2}u(x,y,t)|^2+ |D_x^{-1/2}\partial_y u(x,y,t)|^2-\frac{1}{3}u^3(x,y,t) \, dxdy. \] In the instability literature, the same equation is also written as \[ \partial_t u- \mathcal{R}_1 \Delta u+ \frac{1}{2}\partial_{x_1}(u^2)=0, \] with \(\mathcal R_1=-\partial_{x_1}|\nabla_x|^{-1}\), emphasizing the connection to the fractional operator \(\partial_{x_1}|\nabla_x|\) [1705.01627][2509.14870]. ## 5. Well-posedness, weighted theory, and solitary waves Local well-posedness for the Shrira PDE is established in Sobolev spaces \(H^s(\mathbb K^2)\), \(\mathbb K\in\{\mathbb R,\mathbb T\}\), for \(s>3/2\), and also in the anisotropic energy-adapted space \(\widetilde{X}^s(\mathbb R^2)\) with norm \[ \|f\|_{\widetilde{X}^s}=\|J_x^sf\|_{L^2_{xy}}+\|D_x^{-1/2}\partial_y f\|_{L^2_{xy}}. \] The same framework yields results in weighted anisotropic Sobolev spaces \[ Z_{s,r_1,r_2}(\mathbb{R}^2)=H^{s}(\mathbb{R}^2)\cap L^{2}((| x|^{2r_1}+|y|^{2r_2}) \, dx dy), \] their mean-zero variants \(\dot Z_{s,r_1,r_2}\), and the mixed space \(ZH_{s,1/2,r_2}\). The paper further derives unique-continuation consequences, including the identity \[ 2i\sin(\eta^2(t_2-t_1))\partial_{\xi}\widehat{u}(0,\eta,t_1)=-\int_{t_1}^{t_2}\sin(\eta^2(t_2-t'))\widehat{u^2}(0,\eta,t')\, dt', \] and the corollary that if \(\partial_{\xi}\widehat{u}(0,\eta,t_1)=0\) for a.e. \(\eta\), then \(u\equiv 0\) [2005.09184]. For the generalized equation, solitary waves are sought in the form \[ u(x,y,t)=\varphi(x-ct,y), \qquad c>0, \] which yields \[ -c\varphi_x-\mathscr H\Delta \varphi + (f(\varphi))_x=0, \] or, after integrating in \(x\), \[ -c\varphi-\mathscr H\partial_x^{-1}\Delta \varphi + f(\varphi)=0. \] The natural energy space is \[ \mathscr Z = \left\{ u\in L^2(\mathbb R^2)\; ;\; D_x^{1/2}u\in L^2(\mathbb R^2),\; D_x^{-1/2}u_y\in L^2(\mathbb R^2) \right\}, \] with norm \[ \|u\|_{\mathscr Z} = \left( \int_{\mathbb R^2} \big( c\,u^2 + |D_x^{1/2}u|^2 + |D_x^{-1/2}u_y|^2 \big)\,dx\,dy \right)^{1/2}. \] Under assumptions (A1)–(A4), the stationary problem has a nontrivial solution \(\varphi\in\mathscr Z\), 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 \(W^{1,r}(\mathbb R^2)\) for \(1