We consider a prototypical model for water wave dynamics in intermediate depths and derive linearly well-posed boundary conditions. These boundary conditions allow us to model a rigid wall, a wave generator, a steady-state flow, and wave outflow. To solve the differential equations along with these boundary conditions, we introduce a summation-by-parts finite-difference scheme and prove its linear stability. Finally, we demonstrate high-order convergence in practical computations, validating the robustness of our scheme. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We consider the system of partial differential equations proposed in [1] as an alternative to the Navier-Stokes equations. These two sets of equations differ primarily in that the former incorporates diffusive terms of mass, momentum and energy. While existence of solutions to a weak version of the diffusive system is demonstrated in [1], we further reduce the diffusive differential equations and their weak counterparts using the isentropic assumption. Under specific technical assumptions, we establish a form of uniqueness known as weak-strong uniqueness for the reduced systems. This ensures that a solution to the differential equations and a solution to the weak counterpart are equivalent provided they originate from the same initial data.

This paper investigates the well-posedness of a class of one-dimensional Boussinesq-type systems modeling nonlinear dispersive water waves over varying bottom topographies and offers the first rigorous analytical framework for the initial-value formulation of the Madsen and Sørensen Boussinesq-type equations. The equations under consideration depend on a real parameter B, which modulates the strength of the dispersive effect, and on the static water depth h_0 . In the case of flat bathymetry, we provide a detailed analysis of the associated key operator, formulate the quasilinear system, and construct a suitable symmetrizer to define an appropriate energy norm. Local well-posedness in the "hyperbolic" space is established under three distinct assumptions on the pair (B, h_0) . The analysis is subsequently extended to the variable bathymetry setting in the critical case B = -1/6 , which is the only value for which the model admits a mathematically tractable structure allowing for a rigorous treatment. For this setting, we derive a symmetrization and prove local well-posedness results under natural conditions.

Boussinesq systems, first introduced by J. Boussinesq in 1872, model small-amplitude shallow-water waves with weakly nonlinear and dispersive effects. Over time, various forms of these systems have been developed. A common feature is their solution variables: the time-dependent free-surface elevation and horizontal fluid velocity in the water column. In 1993, O. Nwogu introduced a modified Boussinesq system by taking the solution variable at a specific fluid depth. This innovation significantly improved the linear dispersion properties, extending the applicability of Boussinesq systems to greater water depths. Due to these enhanced properties, the Nwogu system has become a widely used tool for nearshore wave modelling. While Boussinesq systems are commonly applied to small-amplitude long waves, the mechanical properties of these systems- such as mass, momentum, and energy-have received comparatively little attention, even for the Nwogu system. By leveraging the conservation principle, which states that the rate of change of a quantity in a spatial region equals the net influx into that region, mechanical balance laws can be derived for water wave models. This study derives mass, momentum, and energy balance laws for the Nwogu system in a two-dimensional domain with realistic bottom topography, ensuring accuracy consistent with the system's asymptotic precision.

The aim of this paper is to provide an alternative proof of the well-posedness of the Green–Naghdi equations with the Coriolis effect established by Chen et al. (2018). We showed that an additional assumption on the initial horizontal velocity is not necessary to obtain well-posedness. Indeed, with a refined symmetrizer and appropriately scaling the rotation parameter, we can derive a prior energy estimate based solely on the physically relevant depth-condition.

It is shown that the Boussinesq-Peregrine system, which describes long waves of small amplitude at the surface of an inviscid fluid with variable depth, admits a number of approximate conservation equations. Notably, this paper provides accurate estimations for the approximate conservation of the mechanical balance laws associated with mass, momentum, and energy. These precise estimates offer valuable insights into the behavior and dynamics of the system, shedding light on the conservation principles governing the wave motion.

This article takes into account the Korteweg-de Vries (KdV) equation as an approximate model of long waves of small amplitude at the free surface with inviscid fluid. It is demonstrated that the mechanical balance quantities, as defined by the solution of the KdV equation, rigorously approximate those in the Euler system within the L infinity space. Furthermore, these approximations are estimated in relation to the parameter epsilon characterizing the long-wave behavior.

In the context of the initial data and an amplitude parameter ε , we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space H^k as long as k>5/2 . Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of ε ^-1, while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and sech^2 .

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis‐Procesi equation. Journal of Differential Equations . 2020;269(10):7730‐7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr (A new highly nonlinear shallow water wave equation. Journal of Evolution Equations . 2016;16(3):539‐556) are indeed traveling waves, with further information on their steady structures. We also provide a simple proof that symmetric waves are traveling waves to the free surface evolution equation of moderate amplitude waves in shallow water.

We derive the classical Boussinesq system to model equatorial water flows with the weak Coriolis effect due to Earth’s rotation. The system is derived from the Zakharov–Craig–Sulem formulation of the water wave equations in the f-plane approximation. Then, for the obtained system, we find some exact traveling wave solutions of the form sech2 and sech.

This paper is a continuation of a previous work on the extended Green–Naghdi system. We prolong the system, in arbitrary dimension, with/without surface tension, and for a general bottom topography. Confining the work to the one-dimensional case, wellposedness and consistency with respect to initial data and parameters are proved, taking into account the effect of surface tension. The results are local, but long term in the sense of dependence upon initial data. As a conclusion, our solution remains close to the exact solution of the full Euler system with a better (smaller) precision and therefore the full justification of the models.

In the mathematical theory of water waves, this paper focuses on the hierarchy of higher order asymptotic models. The well-posedness of the medium amplitude extended Green-Naghdi model, as well as higher-ordered Boussinesq-Peregrine and Boussinesq models, is first demonstrated. Introducing a regularization term and various physical topography variations, we show that these models admit unique solutions by a standard energy estimate method in the "hyperbolic" space \begin{document}$ H^{s+2}( \mathbb R)^2 $\end{document}, \begin{document}$ s>3/2 $\end{document}, but on short/intermediate time scales with respect to amplitude and topography parameters of order \begin{document}$ \varepsilon^{-1/4} $\end{document}, \begin{document}$ \max(\sqrt{ \varepsilon}, \beta)^{-1} = \beta^{-1} $\end{document} and \begin{document}$ \varepsilon^{-1/2} $\end{document} respectively. Furthermore, we show that the extended Green-Naghdi system's long-term well-posedness is reachable on time scales of order \begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}. The above three specified models, in particular, admit longer time existence of order \begin{document}$ \varepsilon^{-1} $\end{document}, \begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}, and \begin{document}$ \varepsilon^{-1} $\end{document}, respectively.

The purpose of this paper is to present the derivation and mathematical analysis of a new asymptotic model that describes the evolution of medium amplitude internal waves propagating between a flat rigid-lid and a highly variable topography. The smallness assumptions on the topographic variation parameter used in Lteif et al. (2015) [19] and in Lteif and Israwi (2018) [18] are now relaxed and the results of the aforementioned papers are improved and generalized to the complex case of large topography variation. Limiting the flow to one-layer, we also emphasize our model's well-posedness in comparison to the original asymptotic model.

This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order $1/\sqrt{\eps}$ and showing that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.

In this paper, we prove an orbital stability result for the Degasperis-Procesi peakon with respect to perturbations having a momentum density that is first negative and then positive. This leads to the orbital stability of the antipeakon-peakon profile with respect to such perturbations.

The aim of this paper is to give an alternative proof for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long‐term well‐posedness result established by Israwi (Derivation and analysis of a new 2D Green–Naghdi system, Nonlinearity, 2010, 23, 2889–2904) but for the original system, in which the partial operator is not involved.

Green-Naghdi equations are commonly used in coastal oceanography to describe the propagation of large-amplitude surface waves. In this paper, a new convenient equivalent system to the standard two-dimensional case of these equations is presented. This system helps in studying the existence of solution and in its numerical simulations, also gives some physical properties as the irrationality phenomena.

In this paper, a generalized nonlinear Kawahara equation with time and space‐dependent coefficient is considered. We show that the construction of solution with a standard fixed point method can be accomplished so that the well‐posedness in for some is proved under a specific nonpositive differential condition. We introduce an adequate weight function to define the energy and a nondegenerate condition is assumed on the fifth dispersive coefficient.

In this paper, we will derive the two-dimensional extended Green-Naghdi system {see Matsuno [Proc. R. Soc. A 472, 20160127 (2016)] for determination in a various way} for flat bottoms of order three with respect to the shallowness parameter μ. Then we consider the 1D extended Green-Naghdi equations taking into account the effect of small surface tension. We show that the construction of solution with a standard Picard iterative scheme can be accomplished in which the well-posedness in Xs=Hs+2(R)×Hs+2(R) for some s>32 of the new extended 1D system for a finite large time existence t=O(1ε) is demonstrated.