Solitary Water Waves Research Articles

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Korteweg–De Vries–Burger Equation with Jeffreys’ Wind–Wave Interaction: Blow-Up and Breaking of Soliton-like Solutions in Finite Time

In this study, the evolution of surface water solitary waves under the action of Jeffreys’ wind–wave amplification mechanism in shallow water is analytically investigated. The analytic approach is essential for numerical investigations due to the scale of energy dissipation near coasts. Although many works have been conducted based on the Jeffreys’ approach, only some studies have been carried out on finite depth. We show that nonlinearity, dispersion, and anti-dissipation are the dominating phenomena, obeying an anti-diffusive and fully nonlinear Serre–Green–Naghdi (SGN) equation. Applying an appropriate perturbation method, the current research yields a Korteweg–de Vries–Burger-type equation (KdV-B), combining weak nonlinearity, dispersion, and anti-dissipation. This derivation is novel. We show that the continuous transfer of energy from wind to water results in the growth over time of the KdV-B soliton’s amplitude, velocity, acceleration, and energy, while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys’ approach and is due to finite depth. In this study, it is shown that expansion and breaking occur in finite time. These times are calculated and expressed with respect to soliton- and wind-appropriateparameters and values. The obtained values are measurable in experimental facilities. A detailed analysis of the breaking time is conducted with regard to various criteria. By comparing these times to the experimental results, the validity of these criteria are examined.

Large-Amplitude Steady Solitary Water Waves with Constant Vorticity

This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. This is stronger than what one would expect from a straightforward generalization of existing results for periodic waves. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave.

Velocity field and pressure analysis of equatorial flows beneath solitary water waves

In this paper, we investigate steady equatorial flows beneath solitary water waves subject to the Coriolis effect, which propagate over a flat bed. In particular, we focus on irrotational flows and present some properties of velocity field, behavior of the pressure and the extrema of the dynamic pressure. In addition, we provide some estimates for the elevation of wave from pressure measurements at an arbitrary intermediate depth. The analysis is based on the maximum principles.

No Pure Capillary Solitary Waves Exist in 2D Finite Depth

We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth.

Numerical solution of fractional Dullin–Gottwald–Holm equation for solitary shallow water waves

AbstractIn this paper, a numerical scheme to solve the time‐fractional Dullin–Gottwald–Holm equation has been proposed. The proposed scheme is based on approximating the solution using multiquadric radial basis function. The fractional derivatives are described in the Caputo sense. The obtained numerical results are compared with exact solution to confirm the accuracy and effectiveness of the presented scheme.

Solitary water wave interactions for the forced Korteweg–de Vries equation

The aim of this work is to study solitary water wave interactions between two topographic obstacles for the forced Korteweg–de Vries equation (fKdV). Focusing on the details of the interactions, we identify regimes in which solitary wave interactions maintain two well separated crests and regimes where the number of local maxima varies according to the laws \(2\rightarrow 1\rightarrow 2\rightarrow 1\rightarrow 2\) or \(2\rightarrow 1\rightarrow 2\). It shows that the geometric Lax-categorization of Korteweg–de Vries (KdV) two-soliton interactions still holds for the fKdV equation.

Nonexistence of Subcritical Solitary Waves

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.

Solitary water waves created by variations in bathymetry

Abstract

A new definition of the kinematic breaking onset criterion validated with solitary and quasi-regular waves in shallow water

A large body of work has been devoted to the accurate detection and simulation of wave breaking in coastal areas. It is a key process for a wide range of engineering activities and environmental issues. This has motivated the development of a variety of breaking onset criteria, such as kinematic criteria based on a maximum value (usually unity) of the ratio uc/c, of the horizontal particle velocity at the wave crest uc to its phase velocity c, both taken in the direction of wave propagation. Here, we investigate numerically the validity of this criterion in capturing breaking onset for solitary and quasi-regular two-dimensional shallow water waves using the Fully Nonlinear Potential Flow (FNPF) model by Grilli and Subramanya (1996). With this model, the propagation up to overturning of solitary waves over plane slopes, and solitary and quasi-regular waves over a submerged bar, both initially specified as numerically exact FNPF waves, is simulated. In all cases, waves break as spilling or plunging breakers, initiated by the formation of a breaker jet near the wave crest. Results show that the location of the maximum fluid velocity ||u||m on the free surface closely coincides with the location where the overturning jet is initiated. Based on this, a new breaking onset criterion is proposed as ||u||m/c≃1, which is shown to be more universal for accurately detecting wave breaking initiation than existing criteria based on the crest velocity.

On the Stability of Solitary Water Waves with a Point Vortex

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small‐amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite‐dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS) , but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state‐dependent.As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized Korteweg–de Vries (KdV) and Benjamin‐Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss , who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion and obtain a new instability result for fractional KdV. © 2020 Wiley Periodicals, Inc.

Inertial energy dissipation in shallow-water breaking waves

We present direct numerical simulations of breaking solitary waves in shallow water to quantify the energy dissipation during the active breaking time. We find that this dissipation can be predicted by an inertial model based on Taylor’s hypothesis as a function of the local wave height, depth and the beach slope. We obtain a relationship that gives the dissipation rate of a breaking wave on a shallow slope as a function of local breaking parameters. Next, we use empirical relations to relate the local wave parameters to the offshore conditions. This enables the energy dissipation to be predicted in terms of the initial conditions. We obtain good collapse of the numerical data with respect to the theoretical scaling.

Existence, Nonexistence, and Asymptotics of Deep Water Solitary Waves with Localized Vorticity

In this paper, we study solitary waves propagating along the surface of an infinitely deep body of water in two or three dimensions. The waves are acted upon by gravity and capillary effects are allowed --- but not required --- on the interface. We assume that the vorticity is localized in the sense that it satisfies certain moment conditions, and we permit there to be finitely many point vortices in the bulk of the fluid in two dimensions. We also consider a two-fluid model with a vortex sheet. Under mild decay assumptions, we obtain precise asymptotics for the velocity field and free surface, and relate this to global properties of the wave. For instance, we rule out the existence of waves whose free surface elevations have a single sign and of vortex sheets with finite angular momentum. Building on the work of Shatah, Walsh, and Zeng, we also prove the existence of families of two-dimensional capillary-gravity waves with compactly supported vorticity satisfying the above assumptions. For these waves, we further show that the free surface is positive in a neighborhood of infinity, and that the asymptotics at infinity are linked to the net vorticity.

On the Existence and Instability of Solitary Water Waves with a Finite Dipole

This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is vacuum, and that the waves are acted upon by gravity with surface tension effects on the air--water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted $\delta$-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of $0$. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis--Shatah--Strauss method recently introduced by Varholm, Wahl\'en, and Walsh.

Integral and Asymptotic Properties of Solitary Waves in Deep Water

AbstractWe consider two‐ and three‐dimensional gravity and gravity‐capillary solitary water waves in infinite depth. Assuming algebraic decay rates for the free surface and velocity potential, we show that the velocity potential necessarily behaves like a dipole at infinity and obtain a related asymptotic formula for the free surface. We then prove an identity relating the “dipole moment” to the kinetic energy. This implies that the leading‐order terms in the asymptotics are nonvanishing and in particular that the angular momentum is infinite. Lastly we prove that the “excess mass” vanishes. © 2018 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

Solitary water waves with discontinuous vorticity

We investigate the existence of solitary gravity waves traversing a two-dimensional body of water that is bounded below by a flat impenetrable ocean bed and above by a free surface of constant pressure. Our main interest is constructing waves of this form that exhibit a discontinuous distribution of vorticity. More precisely, this means that the velocity limits both upstream and downstream to a laminar flow that is merely Lipschitz continuous. We prove that, for any choice of background velocity with this regularity, there exists a global curve of solutions bifurcating from a critical laminar flow and including waves arbitrarily close to having stagnation points. Each of these waves has an axis of even symmetry, and the height of their streamlines above the bed decreases monotonically as one moves to the right of the crest.

Fully nonlinear capillary–gravity solitary waves under a tangential electric field, Part II: Dynamics

The stability and dynamics of two-dimensional fully nonlinear capillary–gravity solitary waves are studied when a uniform electric field is applied in the direction parallel to the undisturbed free surface of a dielectric fluid. For simplicity, we assume that the permittivity of the gas above the fluid is much smaller, therefore the two-layer problem can be reduced to be a one-layer one. The existence of fully localized solitary waves in deep water has been thoroughly studied in our previous work (Tao and Guo, 2014) via weakly nonlinear normal form analysis and fully nonlinear numerical computation. In the present paper, the stability and dynamics of the obtained solitary waves are investigated based on the time-dependent conformal map technique. Similar to the nonelectric capillary–gravity solitary waves, all depression waves, together with elevation waves featuring two big troughs connected by a small dimple, are found to be stable. Furthermore, head-on and overtaking collisions between stable solitary waves are studied via a numerical time integration.

Solitary water waves propagating on an underlying uniform current over a finite depth

We investigate the nature of velocity and pressure fields for solitary water waves (without any restriction on amplitude) propagating over finite depth with uniform underlying currents. We show that two cases exist corresponding to the fluid velocities u greater than or less than the solitary wave propagation speed c, and these two cases result in different nature of flow fields. We prove the conditions for existence of solitary waves in 2D with uniform underlying current over finite depth for these two cases. We also derive some bounds on Froude number and maximum amplitude of solitary wave height.

Numerical simulation of solitary waves on deep water with constant vorticity

Characteristics of solitary deep water waves on a flow with constant vorticity are investigated by numerical simulation within the framework of fully nonlinear equations of motion (Euler equations) using the method of surface-tracking conformal coordinates. To ensure that solutions observed are stable, soliton formation as a result of disintegration of an initial pulse-like disturbance is modeled. Evidence is obtained that solitary waves with height above a certain threshold are unstable.