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A nonlinear Schr\"odinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can…

We study the problem of the motion of the free surface of a liquid. We prove existence and stability for the linearized equations.

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the…

We announce a detailed investigation of limits of N-soliton solutions of the Korteweg-deVries (KdV) equation as $N$ tends to infinity. Our main results provide new classes of KdV-solutions including in particular new types of soliton-like…

We study a spherical, self-gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity-induced dispersion, the emergence of solitons becomes possible. We…

The two-dimensional evolution of perturbed long weakly-nonlinear surface plane, ring, and hybrid waves, consisting, to leading order, of a part of a ring and two tangent plane waves, is modelled numerically within the scope of the 2D…

We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density…

Surface-subsurface flow models for hydrological applications solve a coupled multiphysics problem. This usually consists of some form of the Richards and shallow water equations. A typical setup couples these two nonlinear partial…

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather…

We study the perturbed Burgers-Korteweg-de Vries equation. This equation can be used for the description of nonlinear waves in a liquid with gas bubbles and for the description of nonlinear waves on a fluid layer flowing down an inclined…

The Korteweg-de Vries (KdV) equation is a non-linear wave equation that has played a fundamental role in diverse branches of mathematical and theoretical physics. In the present paper, we consider its significance to cosmology. It is found…

We consider propagation of solitons along large scale background waves in the generalized Korteweg-de Vries (gKdV) equation theory when the width of the soliton is mach smaller than the characteristic size of the background wave. Due to…

Fluid configurations in three-dimensions, displaying a plausible decay of regularity in a finite time, are suitably built and examined. Vortex rings are the primary ingredients in this study. The full Navier-Stokes system is converted into…

In 1895, Korteweg and de Vries (KdV), derived their celebrated equation describing the motion of waves of long wavelength in shallow water. In doing so they made a number of quite reasonable assumptions, incompressibility of the water and…

The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known…

The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that…

An initial value problem of the one-dimensional nonlinear Schr\"odinger (NLS) equation with constant dispersive and nonlinear coefficients can be solved using a compact finite difference scheme (Xie, Li, & Yi, 2009). A similar scheme is…

Density functional theory is used to study colloidal hard-rod fluids near an individual right-angled wedge or edge as well as near a hard wall which is periodically patterned with rectangular barriers. The Zwanzig model, in which the…

We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not…

In linear science, the wave motion equation with general D'Alembert wave solutions is one of the fundamental models. The D'Alembert wave is an arbitrary travelling wave moving along one direction under a fixed model (material) dependent…