Related papers: Decoupled and unidirectional asymptotic models for…
This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present…
The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it…
We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order…
We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV…
We consider here asymptotic models that describe the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with uneven bottoms. The…
In this paper we derive a higher-order KdV equation (HKdV) as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents…
We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves…
A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on…
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation,…
We derived here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid…
The Cauchy problem for the Burgers equation with a small dissipation and an initial weak discontinuity and the Cauchy problem with a large initial gradient for a quasilinear parabolic equation and for the Korteweg-de Vries (KdV) equation…
It is shown that the emergence of obstacles to asymptotic integrability in the analysis of perturbed evolution equations may, often, be a consequence of the manner, in which the freedom in the ex-pansion is exploited in the derivation of…
We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one dimensional waves, and consider the case of a flat bottom. Starting from the…
The propagation of water waves of finite depth and flat bottom is studied in the case when the depth is not small in comparison to the wavelength. This propagation regime is complementary to the long-wave regime described by the famous KdV…
We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of…
The anisotropy of many one-dimensional and first-order-in-time (T$^1$) scalar wave equations (e.g., Korteweg-de Vries and Camassa-Holm) limits their physical completeness and applicability to bidirectional/high-dimensional systems. We…
Considered in this paper is a bi-directional model for the propagation of interfacial capillary-gravity waves in a two-layer system of fluids with rigid lid condition for the upper layer and lower layer with a much larger or infinite depth.…
Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich's f-plane extension of the Miyata-Maltseva-Choi-Camassa (MMCC) model. Within the scope of this model, we develop an asymptotic procedure which…
We consider travelling internal waves in a two-layer fluid with linear shear currents from the viewpoint of the extended Korteweg-de Vries (eKdV) equation derived from a strongly-nonlinear long-wave model. Using an asymptotic…
In this paper we introduce a new reformulation of the Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves over a flat topography derived by Duch\^ene, Israwi and Talhouk. These new Green-Naghdi systems are…