Department of Mathematical Sciences
Tel: +44 (0) 1509 22 2861
Fax: +44 (0) 1509 22 3969
Dr Karima Khusnutdinova
Reader in Applied MathematicsDepartment of Mathematical Sciences
Loughborough, Leicestershire, LE11 3TU
Telephone: +44 (0)1509 228202
Fax: +44 (0)1509 223969
Office: W 2.06
Thanks to the LMS Publications Catalogue for the photo.
Most of the recent research was in the area of nonlinear bulk strain waves in layered solid waveguides. Initially supported by an EPSRC grant, it involved analytical, numerical and experimental studies. The latter were performed by a subcontracted experimental group in the Ioffe Institute of the Russian Academy of Sciences in St. Petersburg. In particular, jointly with Prof. A.M. Samsonov we studied the scattering of a long longitudinal bulk strain solitary wave in a split layered bar made of a hyperelastic (Murnaghan) material (Phys. Rev. E 77 (2008) 066603). The problem was reduced to finding a solution to a Boussinesq-type nonlinear partial differential equation with piecewise-constant coefficients, subject to some continuity conditions across the jump and some natural radiation conditions. The developed analytical approach allowed us to show that splitting (delamination) in a layered structure induces a generation of a train of secondary solitary waves from a single incident soliton and describe some more subtle higher order effects, which could find useful applications in nondestructive testing of layered structures and seismology. The theory was confirmed by experiments (J. Appl. Phys. 107 (2010) 034909), and our analytical approach can be used to solve other similar problems. A system of coupled Bossinesq-type equations was derived as an accurate asymptotic model describing long nonlinear longitudinal waves in layered waveguides with a sufficiently soft bonding layer (Phys. Rev. E 79 (2009) 056606). The explicit weakly nonlinear solution of the initial-value problem for Boussinesq-type equations and systems was constructed in terms of the asymptotically exact (KdV-like) models for unidirectional waves (see Wave Motion 48 (2011) 738-752 and IMA J. Appl. Math. 77 (2012) 361-381).
Some other recent research was devoted to the mutual effects of bubbles and internal waves in the ocean. Bubbles are injected by breaking surface waves, and for strong winds there is a continuous bubble layer of variable thickness which extends to several meters beneath the surface. Jointly with Prof. R.H.J. Grimshaw and Prof. L.A. Ostrovsky we have shown that depth-dependent bubble distributions can support their own "bubble" modes of internal waves, even in an otherwise homogeneous fluid (Eur. J. Mech. B/Fluids 27 (2008) 24-41). Also, bubble layer approximately copies the shape of the internal wave, with a shift in the direction of the propagation of the internal wave, which is important for the study of underwater acoustics and the related oceanographic measurement techniques (Phys. Fluids 22 (2010) 106603).
Most recent paper was devoted to the derivation and study of a new version of the Kadomtsev-Petviashvili equation, related to the elliptic cylindrical geometry and generalising the cKP equation (Chaos 23 (2013) 013126). 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.
A series of joint publications with Prof. E.V. Ferapontov were devoted to the study of special reductions of multi-dimensional quasi-linear systems, describing nonlinear interactions of planar simple waves (Comm. Math. Phys. 248 (2004) 187-206). Quasi-linear systems arise naturally in continuum mechanics. Solutions describing nonlinear interactions of N planar simple waves can be viewed as natural dispersionless analogues of N - soliton solutions, and their existence for N greater than or equal to 3 can be used to classify integrable quasi-linear systems. Double waves (N=2) can exist in non-integrable systems, and such solutions were known in gas dynamics. The existence of double waves was shown to be equivalent to the diagonalizability of an arbitrary matrix of a certain two-parameter family, which can be verified by calculating the Haantjes tensor (Proc. Royal Soc. A 462 (2006) 1197-1219).
Nonlinear Waves in Fluids, 12-14 September 2012, Loughborough University, UK
(dedicated to Roger Grimshaw on the occasion of his retirement)
MAGIC021: Nonlinear Waves (with Prof. R.H.J. Grimshaw and Dr. G. El). Visit MAGIC
MAB170: Probability Theory. Visit MAB170 on Learn Server
MAB250: ODEs and Calculus of Variations. Visit MAB250 on Learn Server
MAC149: Mathematical Methods for Differential Equations (2002 - 2011)
Assistance in the Mathematics Learning Support Centre (2003 - 2006)
2010 - 2012 Placement / Study Leave Tutor
2009 - 2012 Science Faculty Board Representative
2006 - 2010 Programme Tutor for Mathematics BSc and MMath
2005 - 2006 Undergraduate Admissions Tutor (International Students)