In this paper, we present a method for the construction of a robust observer-based H∞ controller for an uncertain time-delay system. Cases of both single and multiple delays are considered. The parameter uncertainties are time-varying and norm-bounded. Observer and controller are designed to be such that the uncertain system is stable and a disturbance attenuation is guaranteed, regardless of the uncertainties. It has been shown that the above problem can be solved in terms of two linear matrix inequalities (LMIs). Finally, an illustrative example is given to show the effectiveness of the proposed techniques.

The authors derive a general theorem on partly bilateral and partly unilateral generating functions involving multiple series with essentially arbitrary coefficients. By appropriately specialising these coefficients, a number of (known or new) results are shown to follow as applications of the theorem.

Solitary wave interaction is examined using an extended Benjamin-Bona-Mahony (eBBM) equation. This equation includes higher-order nonlinear and dispersive effects and is is asymptotically equivalent to the extended Korteweg-de Vries (eKdV) equation. The eBBM formulation is preferable to the eKdV equation for the numerical modelling of solitary wave collisions, due to the stability of its finite-difference scheme. In particular, it allows the interaction of steep waves to be modelled, which due to numerical instability, is not possible using the eKdV equation.

Numerical simulations of a number of solitary wave collisions of varying nonlinearity are performed for two special cases corresponding to surface water waves. The mass and energy of the dispersive wavetrain generated by the inelastic collision is tabulated and used to show that the change in solitary wave amplitude after interaction is of O(α4), verifying previously obtained theoretical predictions.

We study the propagation of electromagnetic waves (EMWs) in both isotropic and anisotropic ferromagnetic material media. As the EMW propagates through linear charge-free isotropic and anisotropic ferromagnetic media, it is found that the magnetic field and the magnetic induction components of the EMW and the magnetization excitations of the medium are in the form of solitons. However, the electromagnetic soliton gets damped and decelerates in the case of a charged medium. In the case of a charge-free nonlinear ferromagnetic medium we obtain results similar to those for the linear case.

We discuss Ablowitz-Segur's connection problem for the second Painlevé equation from the viewpoint of WKB analysis of Painlevé transcendents with a large parameter. The formula they first discovered is rederived from a suitable combination of connection formulas for the first Painlevé equation.

A hierarchy of bilinear Lotka-Volterra equations with a unified structure is proposed. The bilinear Bäcklund transformation for this hierarchy and the corresponding canonical Lax pair are obtained. Furthermore, the nonlinear superposition formula is proved rigorously.

In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2 × 2 Lax matrices, have 2N degrees of freedom. Currently existing methods only enable Lax matrices to yield the first N pairs of canonical separated variables. An approach for constructing the second N pairs of canonical separated variables with N additional separated equations is introduced. The Jacobi inversion problems for binary constrained flows are then established. Finally, the separability of binary constrained flows together with the factorization of soliton equations by the spatial and temporal binary constrained flows leads to the Jacobi inversion problems for soliton equations.

It is shown that an integrable class of helicoidal surfaces in Euclidean space E3 is governed by the Painlevé V equation with four arbitrary parameters. A connection with sphere congruences is exploited to construct in a purely geometric manner an associated Bäcklund transformation.

Using the Bäcklund transformations for the solutions of fourth-order differential equations of the P2 and K2 hierarchies, corresponding discrete equations are found.

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schrödinger operator and hence construct a Darboux transformation, an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discuss the specialisation to the case of a graph with trivial compact part, that is, with diagonal matrix potential.

In this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.