Water waves over a variable bottom: a non-local formulation and conformal mappings

Slowly modulated water waves are considered in the presence of a strongly disordered bathymetry. Previous work is extended to the case where the random bottom irregularities are not smooth and are allowed to be of large amplitude. Through the combination of a conformal mapping and a multiple‐scales asymptotic analysis it is shown that large variations of a disordered bathymetry can affect the nonlinearity coefficient of the resulting damped nonlinear Schrödinger equations. In particular it is shown that as the bathymetry fluctuation level increases the critical point (separating the focusing from the defocusing region) moves to the right, hence enlarging the region where the dynamics is of a defocusing character.

| J | (ξ, t) = z_{ξ}^{2} + z_{ζ}^{2} .

Section: Surface Wave Equations and Conformal Mappingmentioning

confidence: 99%

“…This simpler relation is a consequence of defining a symmetric Z -plane domain, as depicted earlier. Details of the numerical conformal mapping formulation can be found in Fokas and Nachbin [11]. Using the Cauchy-Riemann equations it is easy to note that along the free surface…”

Section: Surface Wave Equations and Conformal Mappingmentioning

confidence: 99%

Wave Packet Defocusing Due to a Highly Disordered Bathymetry

Luz

Nachbin

2013

“…This includes the analysis of BVPs with distributional data and corner singularities [10]. (v) The new method can be applied to linear PDEs with nonlinear boundary conditions, see, for example [11][12][13]. (vi) The first steps have been taken toward extending the unified transform to three dimensions, see, for example [11,14].…”

Section: Introductionmentioning

confidence: 99%

Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

Hitzazis

Fokas

2017

Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary value problem (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late 1990s. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three-dimensional cylindrical domain with a polygonal cross-section.

“…However, KP-I is a physical model appropriate for fluids with strong surface tension making it an inapplicable model for water wave motion. In [21], very general shallow water equations over varying bathymetry were derived but for steady state bathymetries. Further, [21] did not examine the details of the propagation of three-dimensional waves.…”

Section: Introductionmentioning

confidence: 99%

“…In [21], very general shallow water equations over varying bathymetry were derived but for steady state bathymetries. Further, [21] did not examine the details of the propagation of three-dimensional waves. In [22], an fBL equation was used to describe oblique waves, yet no systematic derivation and scale classification were given.…”

Section: Introductionmentioning

confidence: 99%

Three‐Dimensional Surface Water Waves Governed by the Forced Benney–Luke Equation

Curtis

Shen

2015

This paper studies the propagation of three‐dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream‐radiating three‐dimensional solitons are observed, which are different from the upstream‐radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.