Karima R. Khusnutdinova

Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable. The most important examples include the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics and its six-dimensional generalization arising in the context of sdiff(Σ2) self-dual Yang–Mills equations. Given a multidimensional PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions. As an example of this phenomenon we discuss the second commuting flow of the dispersionless KP hierarchy. Considered separately, this is a four-dimensional PDE which does not pass the integrability test. However, the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full (2+1)-dimensional dis...

On the exchange of energy in coupled Klein-Gordon equations

Abstract We consider a system of coupled Klein–Gordon equations, which models one-dimensional nonlinear wave processes in two-component media. We find both linear and nonlinear solutions involving the exchange of energy between the different components of the system. The solutions are a continuum generalization of the classical example of energy exchange in Mandelshtam’s system of coupled pendulums.

Splitting induced generation of soliton trains in layered waveguides

We report first experimental registration of the splitting induced generation of a soliton train from a single incident strain soliton in two- and three-layered elastic waveguides. The origin is in the nonlinear response of the wave to an abrupt change of physical properties of the waveguide. We show a good agreement between our experimental results and theoretical estimates, based on a weakly nonlinear solution for the doubly dispersive (Boussinesq type) equation with piecewise constant coefficients for the waveguide made of a piecewise isotropic nonlinearly elastic material.

Comparison of the effect of cyanoacrylate- and polyurethane-based adhesives on a longitudinal strain solitary wave in layered polymethylmethacrylate waveguides

We study the effect of two types of adhesive bonding on the propagation of a localized longitudinal strain wave in two- and three-layered elastic bars. The detectable variation in the decay rate of the wave at relatively long distances of propagation is observed. It is proposed that such variation can be used as an indicator of the type of an interface.

Bulk strain solitary waves in bonded layered polymeric bars with delamination

We report the registration of delamination induced variations in the dynamics of bulk strain solitary waves in layered polymeric bars with the glassy and rubber-like adhesives, for the layers made of the same material. The key phenomenon in a layered structure with the glassy bonding is the delamination caused fission of a single incident soliton into a wave train of solitons, with the detectable increase in the amplitude of the leading solitary wave. The significant feature of bulk strain solitons in structures bonded with the rubber-like adhesive is the generation of radiating solitary waves, whilst co-propagating ripples disappear in the delaminated area. The observed variations may be used for the detection of delamination in lengthy layered structures.

On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations

In the weakly nonlinear limit, oceanic internal solitary waves for a single linear long wave mode are described by the KdV equation, extended to the Ostrovsky equation in the presence of background rotation. In this paper we consider the scenario when two different linear long wave modes have nearly coincident phase speeds and show that the appropriate model is a system of two coupled Ostrovsky equations. These are systematically derived for a density-stratified ocean. Some preliminary numerical simulations are reported which show that, in the generic case, initial solitary-like waves are destroyed and replaced by two coupled nonlinear wave packets, being the counterpart of the same phenomenon in the single Ostrovsky equation.

Coupled Ostrovsky equations for internal waves in a shear flow

In the context of fluid flows, the coupled Ostrovsky equations arise when two distinct linear long wave modes have nearly coincident phase speeds in the presence of background rotation. In this paper, nonlinear waves in a stratified fluid in the presence of shear flow are investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations. The dispersion relation of the system, based on a three-layer model of a stratified shear flow, reveals various dynamical behaviours, including the existence of unsteady and steady envelope wave packets.

The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type,is developed. We prove that the existence of special solutions known as ‘double waves’ is equivalent to the diagonalizability of an arbitrary matrix of the two-parameter familySince the diagonalizability can be effectively verified by calculating the Haantjes tensor, this provides a simple necessary condition for integrability.

On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation.

There exist two versions of the Kadomtsev-Petviashvili (KP) equation, related to the Cartesian and cylindrical geometries of the waves. In this paper, we derive and study a new version, related to the elliptic cylindrical geometry. The derivation is given in the context of surface waves, but the derived equation is a universal integrable model applicable to generic weakly nonlinear weakly dispersive waves. We also show that there exist nontrivial transformations between all three versions of the KP equation associated with the physical problem formulation, and use them to obtain new classes of approximate solutions for water waves.

Modelling of nonlinear wave scattering in a delaminated elastic bar

Integrity of layered structures, extensively used in modern industry, strongly depends on the quality of their interfaces; poor adhesion or delamination can lead to a failure of the structure. Can nonlinear waves help us to control the quality of layered structures? In this paper, we numerically model the dynamics of a long longitudinal strain solitary wave in a split, symmetric layered bar. The recently developed analytical approach, based on matching two asymptotic multiple-scales expansions and the integrability theory of the Korteweg–de Vries equation by the inverse scattering transform, is used to develop an effective semi-analytical numerical approach for these types of problems. We also employ a direct finite-difference method and compare the numerical results with each other, and with the analytical predictions. The numerical modelling confirms that delamination causes fission of an incident solitary wave and, thus, can be used to detect the defect.