Experimental and theoretical work is presented on the control of wave packet dynamics in atomic and molecular systems. Using sequences of phase-locked optical pulses, the link between optical phase and quantum mechanical phase is explored in the Na atom and Naj dimer, representing a step towards the logical engineering of quantum states in more complicated systems. A novel apparatus, constructed to study and control the vibrational dynamics of vibrational wave packets on the Na ionic potential surface, is described in chapter 3. Theoretical simulations for proposed experiment are presented in chapter 4. Control is achieved by exploiting the phase-evolution of the constituent vibrational quantum states within the wave packet superposition. The phase relationship and the accumulated phase difference between the various components of the wave packet is determined, and a sequence of phase-locked optical pulses is employed to selectively enhance or depopulate specific vibrational states, or sets of vibrational states. The quantum state composition of the resulting wave packet, and the efficiency of the control scheme, is determined by calculating the multi-pulse response of the time-dependent vibrational state populations. In chapter 5, the quantum interference between Rydberg electron wave packets in the Na atom is investigated using pairs of phase-locked wave packets, allowing manipulation of the total orbital angular momentum of Na Rydberg atoms. Initially the wave packet is composed of a superposition of s and d Rydberg series. Exploitation of the difference between the quantum defects of the two series allows specific angular momentum compositions within the resultant wave packet to be engineered. Experimentally, this final quantum state distribution is analysed in the frequency domain using state selective field ionisation, and in the time domain using the optical Ramsey method. The theoretical calculations show how the phase difference between pairs of optical pulses is linked to the corresponding Rydberg frequency spectrum, therefore enabling the control of the quantum state composition of the wave packets.

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In this thesis we study the existence of travelling wave type solutions for a reaction diffusion equation in R2 with a nonlinearity which depends periodically on the spatial variable. Specifically we will consider a particular class of nonlinearities where we treat the coefficient of the linear term as a parameter. For this class of nonlinearities we formulate the problem as a spatial dynamical system and use a centre manifold reduction to find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. We then consider what happens if the parameter and the wave speed vary close to zero; by analysing the bifurcations in this case we are able to find travelling wave solutions with periodic and homoclinic structures. Finally we examine what happens to the travelling wave solutions as the period of the periodic dependence in the nonlinearity tends to zero.

This thesis is concerned with the existence and dynamics of travelling solitary waves in lattice equations, specifically a number of models of the discrete nonlinear Schrodinger equation (DNLS). The DNLS occurs in various forms when modelling a wide range of physical processes involving wave propagation. We provide a review of the literature and introduce some of the concepts that will be use to analyse the differential advance-delay equations which occur when posing lattice equations in a travelling frame. To show the existence of travelling solitary wave solutions to the DNLS three main methods are used, namely the pseudo-spectral method, Melnikov's method for the existence of homoclinic orbits and computation of the so-called Stokes constant for a beyond-all-orders asymptotic expansion. The pseudo-spectral method transforms the differential advance-delay equation into a large system of coupled algebraic equations which are solved numerically.