arXiv subjects

Karima Khusnutdinova

Publications and source records attributed to Karima Khusnutdinova.

8 recordsLinked to original sources

arXiv2026

Solitary waves of moderate amplitude and dispersive radiation in the Serre equations: the extended KdV-Whitham approximation

We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves and use it to describe the evolution of initial conditions generating solitary waves with and without significant dispersive radiation, as well as cases of pure dispersive radiation without any solitary waves. In the slow time formulation for the modelled solutions this equation also generates fast propagating resonant forward radiation due to the non-convexity of its linear dispersion curve, which is not present in the direct numerical simulations of the strongly nonlinear Serre parent system (also known as the Su-Gardner and Green-Naghdi equations). We show that the extended KdV-Whitham approximation and the slow space formulation of the eKdV equation are suitable regularisations of the eKdV equation in several cases of interest. Importantly, unlike the KdV-type equations, it can be used to model waves of moderate amplitude. Numerical comparisons are made between the Serre system and several respective reduced models, where simulations are initiated with an approximate soliton solution of the eKdV equation, constructed by use of Kodama-Fokas-Liu near-identity transformation to the KdV equation, as well as a generic localised initial condition.

nlin.PS↗

arXiv2026

Boussinesq-Klein-Gordon and Ostrovsky equations: evolution of cnoidal waves with local defects

The Boussinesq-Klein-Gordon (BKG) equation has emerged in the studies of nonlinear bulk strain waves in layered solid waveguides. The developed bi-directional weakly-nonlinear solution leads to two copies of the Ostrovsky equation, for the right- and left-propagating waves. Importantly, the derivation avoids the so-called `zero-mean contradiction' between the type of initial conditions in the parent equation and in the reduced model. In this paper, we apply the solution to describe the evolution of cnoidal waves with local periodicity defects and generic localised perturbations, and compare the results with the direct numerical simulations for the full BKG equation. The cnoidal waves with the periodicity defects discussed in our work constitute generalised travelling waves of the Korteweg-de Vries equation, while the Ostrovsky equation leads to a strong burst (and may lead to a rogue wave), qualitatively similar to the wavepacket emerging from a soliton initial condition, but appearing much faster. We compare the weakly-nonlinear solution with the direct numerical simulations within the bi-directional setting of the BKG equation and show that the discussed uni-directional waves and evolution scenarios remain stable in the presence of counter-propagating perturbations.

nlin.PS↗

arXiv2025

Evolution of perturbed long nonlinear plane, ring and hybrid surface waves

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 Boussinesq-Peregrine system. Numerical runs are initiated and interpreted using the reduced 2+1-dimensional cKdV-type and KPII equations. The cKdV-type equation leads to two different models, the KdV$θ$ and cKdV equations, depending on whether we use the general or singular (i.e. the envelope of the general) solution of the associated nonlinear first-order differential equation. The KdV$θ$ equation is also derived directly from the 2D Boussinesq-Peregrine system and used to analytically describe the intermediate 2D asymptotics of line solitons subject to sufficiently long transverse perturbations of finite strength, while the cKdV equation is used to initiate outward- and inward-propagating ring waves with localised and periodic perturbations. Both of these equations, together with the KPII equation, are used to model the evolution of hybrid waves, where we show, in particular, that large localised waves (lumps) can appear as transient (emerging and then disappearing) states in the evolution of inward-propagating waves, contributing to the possible mechanisms for the generation of rogue waves. Detailed comparisons are made between the key features of the non-stationary two-dimensional modelling and relevant predictions of the reduced equations.

physics.flu-dyn↗

arXiv2025

Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation

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 allows us to obtain a description of a large class of uni-directional waves leading to the Ostrovsky equation and allowing for the presence of shear inertial oscillations and barotropic transport. Importantly, unlike the conventional derivations leading to the Ostrovsky equation, the constructed solutions do not impose the zero-mean constraint on the initial conditions for any variable in the problem formulation. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg-de Vries (KdV) equation but having a local amplitude and/or periodicity defect, and show that such initial conditions can lead to the emergence of bursts of large internal waves and shear currents. As a by-product of our study, we show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass-Erdmann conditions for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation changes this behaviour and leads to the emergence of strong bursts.

physics.flu-dyn↗

arXiv2025

Internal solitary and cnoidal waves of moderate amplitude in a two-layer fluid: the extended KdV equation approximation

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 Kodama-Fokas-Liu near-identity transformation, we map the eKdV equation to the Gardner equation. This improved Gardner equation has a different cubic nonlinearity coefficient and an additional transport term compared to the frequently used truncated Gardner equation. We then construct approximate solitary and cnoidal wave solutions of the eKdV equation using this mapping and test validity and performance of these approximations, as well as performance of the truncated and improved Gardner and eKdV equations, by comparison with direct numerical simulations of the strongly-nonlinear two-layer long-wave parent system in the absence of currents.

physics.flu-dyn↗

arXiv2023

Nonlinear concentric water waves of moderate amplitude

We consider the outward-propagating nonlinear concentric water waves within the scope of the 2D Boussinesq system. The problem is axisymmetric, and we derive the slow radius versions of the cylindrical Korteweg - de Vries (cKdV) and extended cKdV (ecKdV) models. Numerical runs are initially performed using the full axisymmetric Boussinesq system. At some distance away from the origin, we use the numerical solution of the Boussinesq system as the "initial condition" for the derived cKdV and ecKdV models. We then compare the evolution of the waves as described by both reduced models and the direct numerical simulations of the axisymmetric Boussinesq system. The main conclusion of the paper is that the extended cKdV model provides a much more accurate description of the waves and extends the range of validity of the weakly-nonlinear modelling to the waves of moderate amplitude.

physics.flu-dyn↗

arXiv2021

Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear flow. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive them from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (e.g. group speed) and to constructing more general `hybrid solutions' consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. The detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions of the surface waves in a homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. A global and local measure of the deformation of wavefronts are introduced and evaluated.

physics.flu-dyn↗

arXiv2020

Long internal ring waves in a two-layer fluid with an upper-layer current

We consider a two-layer fluid with a depth-dependent upper-layer current (e.g. a river inflow, an exchange flow in a strait, or a wind-generated current). In the rigid-lid approximation, we find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring wave in different directions in terms of the hypergeometric function. This allows us to obtain an analytical description of the wavefronts and vertical structure of the ring waves for a large family of the current profiles and to illustrate their dependence on the density jump and the type and the strength of the current. In the limiting case of a constant upper-layer current we obtain a 2D ring waves' analogue of the long-wave instability criterion for plane interfacial waves. On physical level, the presence of instability for a sufficiently strong current manifests itself already in the stable regime in the squeezing of the wavefront of the interfacial ring wave in the direction of the current. We show that similar phenomenon can also take place for other, depth-dependent currents in the family.

physics.flu-dyn↗