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Hamiltonian Approach To Modelling Interfacial Internal Waves Over Variable Bottom, Rossen Ivanov, Calin Martin, Michail Todorov
Hamiltonian Approach To Modelling Interfacial Internal Waves Over Variable Bottom, Rossen Ivanov, Calin Martin, Michail Todorov
Articles
We study the effects of an uneven bottom on the internal wave propagation in the presence of stratification and underlying non-uniform currents. Thus, the presented models incorporate vorticity (wave–current interactions), geophysical effects (Coriolis force) and a variable bathymetry. An example of the physical situation described above is well illustrated by the equatorial internal waves in the presence of the Equatorial Undercurrent (EUC). We find that the interface (physically coinciding with the thermocline and the pycnocline) satisfies in the long wave approximation a KdV–mKdV type equation with variable coefficients. The soliton propagation over variable depth leads to effects such as soliton …
On The Intermediate Long Wave Propagation For Internal Waves In The Presence Of Currents, Joseph Cullen, Rossen Ivanov
On The Intermediate Long Wave Propagation For Internal Waves In The Presence Of Currents, Joseph Cullen, Rossen Ivanov
Articles
A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface approximation is assumed. The fluids are incompressible and inviscid. The model equations are obtained from the Hamiltonian formulation of the dynamics in the presence of a depth-varying current. It is shown that an appropriate scaling leads to the integrable Intermediate Long Wave Equation (ILWE). Two limits of the ILWE leading to the integrable Benjamin-Ono and KdV equations are …
Camassa-Holm Cuspons, Solitons And Their Interactions Via The Dressing Method, Rossen Ivanov, Tony Lyons, Nigel Orr
Camassa-Holm Cuspons, Solitons And Their Interactions Via The Dressing Method, Rossen Ivanov, Tony Lyons, Nigel Orr
Articles
A dressing method is applied to a matrix Lax pair for the Camassa–Holm equation, thereby allowing for the construction of several global solutions of the system. In particular, solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa–Holm equation are re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.
Riemann-Hilbert Problem, Integrability And Reductions, Vladimir Gerdjikov, Rossen Ivanov, Aleksander Stefanov
Riemann-Hilbert Problem, Integrability And Reductions, Vladimir Gerdjikov, Rossen Ivanov, Aleksander Stefanov
Articles
Abstract. The present paper is dedicated to integrable models with Mikhailov reduction groups GR ≃ Dh. Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the GR-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with Dh symmetries are presented.
On The N-Wave Equations With Pt-Symmetry, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov
On The N-Wave Equations With Pt-Symmetry, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov
Articles
We study extensions of N-wave systems with PT-symmetry. The types of (nonlocal) reductions leading to integrable equations invariant with respect to P- (spatial reflection) and T- (time reversal) symmetries is described. The corresponding constraints on the fundamental analytic solutions and the scattering data are derived. Based on examples of 3-wave (related to the algebra sl(3,C)) and 4-wave (related to the algebra so(5,C)) systems, the properties of different types of 1- and 2-soliton solutions are discussed. It is shown that the PT symmetric 3-wave equations may have regular multi-soliton solutions for some specific choices of their parameters.
On The Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev
On The Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev
Articles
We develop the direct scattering problem for quadratic bundles associated to Hermitian symmetric spaces. We adapt the dressing method for quadratic bundles which allows us to find special solutions to multicomponent derivative Schrodinger equation for instance. The latter is an infinite dimensional Hamiltonian system possessing infinite number of integrals of motion. We demonstrate how one can derive them by block diagonalization of the corresponding Lax pair.
G-Strands, Darryl Holm, Rossen Ivanov, James Percival
G-Strands, Darryl Holm, Rossen Ivanov, James Percival
Articles
A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) …
Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov
Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov
Articles
The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T∗(TN) to the flow momentum (X∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The …
On The Integrability Of A Class Of Nonlinear Dispersive Wave Equations, Rossen Ivanov
On The Integrability Of A Class Of Nonlinear Dispersive Wave Equations, Rossen Ivanov
Articles
We investigate the integrability of a class of 1+1 dimensional models describing nonlinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.