Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Abstract

Localized perturbations of internal gravity wave fields constitute an integral and important component of nonlinear wave motions of a stratified fluid. It is important to analyze these perturbations because these waves often carry significant amounts of energy and play the leading role in processes in the environ ment. In particular, they affect the transport of bottom sediments in the shelf regions of the oceans, propaga tion of pollutants and impurities in the bulk of the fluid and on its surface, mixing of water, energy dissipation, transfer of acoustic signals, etc. It is impossible to understand the mechanisms of this action without detailed studies of the properties of isolated long lived pulses, accurate determination of their characteristics and features of the dynamics under various conditions in the medium, and the existence of localized nonra diating waveforms depending on the external factors of the medium.The modified Korteweg-de Vries equation with cubic nonlinearity is applicable to many physical problems with symmetry. Anisotropy for waves in a fluid is associated with the direction of the gravita tional force. For this reason, the classical Kortewegde Vries equation, which was first derived for waves on water, is derived in the first approximation of perturba tion theory for weakly nonlinear and weakly dispersive waves in a fluid. At the same time, the structure of the wave field in a stratified fluid is determined by the ver tical distribution of the density field, more precisely, by the distribution of the buoyancy (Brunt-Väisälä) frequency. This distribution can be symmetric; in par ticular, it can include two identical pycnoclines (regions with a strong change in the density). This case is described by the modified Korteweg-de Vries equa tion [1]. Moreover, cubic nonlinearity for certain posi tions of pycnoclines becomes anomalously small and even vanishes [1]. In this case, terms of the next per turbation order should be taken into account. Such an extended modified Korteweg-de Vries equation was recently derived in [2]. This equation is not integrable and the dynamics of solitons of this equation is ana lyzed in this work.First, we determine the situation described by this equation. We consider a three layer (in density) water flow in the gravitational field. The flow is bounded in the vertical direction by the bottom and free surface. The latter can be replaced by a rigid surface (this approximation in oceanology is called the rigid lid condition and is valid at small variations in the density, which are typical of incompressible stratified fluids). Let h be the thickness of the lower and upper layers; H be the total depth of the fluid; ρ 1 = ρ, ρ 2 = ρ + Δρ, and ρ 3 = ρ 2 + Δρ = ρ + 2Δρ be the densities of the lower, middle, and upper layers, respectively; and η(x, t) and ζ(x, t) be the lower and upper interfaces, respectively. We assume that the relative density changes at the Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the co...

Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

“…Note that a similar equation arose in [34] when simulating solitary waves in a three-layer symmetrically stratified fluid in the framework of an extended nonintegrable version of the modified Korteweg-de Vries equation.…”

Some features of the solving of hydrodynamic equations for solitary waves in the open water channel

“…Since the first observation of solitary waves in a water channel by D.S. Russell in 1834 who accompanied a solitary wave on a horse, many different approaches arose in the theory of describing such waves [1][2][3], including for various external conditions [4][5][6]. For example, simulation of solitary waves or solitons in [1] made it possible to describe waves propagating both from left to right and from right to left.…”

Dispersive focusing in fractional Korteweg–de Vries-type equations

“…Completely integrable equations of this type are given, for instance, by the classical KdV equation (m = 2, n = 1), modified KdV equation (m = 2, n = 2), integral Benjamin-Ono equation (m = 1, n = 1), [7]. We would like to mention that many non-intergable equations belonging to this class, such as classical (m = 4, n = 1) and modified (m = 4, n = 2) Kawahara equations, [8][9][10]; Shamel equation (m = 2, n = 1/2), [11]; (2 + 4)-KdV equation (m = 2, n = 4), [13]; reduced Whitham equation for short gravity waves (m = 1/2, n = 1), [12]; reduced Ostrovsky equation (m = −2, n = 1 or 2), [14]. In addition, some studies are available where the equation ( 1) is regarded with negative nonlinearities, e.g.…”