Antonmaria A. Minzoni

Evolution of lump solutions for the KP equation

Abstract The two (space)-dimensional generalisation of the Korteweg-de Vries (KdV) equation is the Kadomtsev-Petviashvili (KP) equation. This equation possesses two solitary wave type solutions. One is independent of the direction orthogonal to the direction of propagation and is the soliton solution of the KdV equation extended to two space dimensions. The other is a true two-dimensional solitary wave solution which decays to zero in all space directions. It is this second solitary wave solution which is considered in the present work. It is known that the KP equation admits an inverse scattering solution. However this solution only applies for initial conditions which decay at infinity faster than the reciprocal distance from the origin. To study the evolution of a lump-like initial condition, a group velocity argument is used to determine the direction of propagation of the linear dispersive radiation generated as the lump evolves. Using this information combined with conservation equations and a suitable trial function, approximate ODEs governing the evolution of the isolated pulse are derived. These pulse solutions have a similar form to the pulse solitary wave solution of the KP equation, but with varying parameters. It is found that the pulse solitary wave solutions of the KP equation are asymptotically stable, and that depending on the initial conditions, the pulse either decays to a pulse of lower amplitude (shedding mass) or narrows down (shedding mass) to a pulse of higher amplitude. The solutions of the approximate ODEs for the pulse evolution are compared with full numerical solutions of the KP equation and good agreement is found.

Spatial soliton evolution in nematic liquid crystals in the nonlinear local regime

Spatial solitons in nematic liquid crystals are considered in the regime of local response of the crystal, such solitons having been called nematicons in previous experimental studies. In the limit of low light intensity and local material response, it is shown that the full governing equations reduce to a single, higher-order nonlinear Schrodinger equation. Modulation equations are derived for the evolution of a nematiconlike pulse; these equations also include the dispersive radiation shed as the pulse evolves. The modulation equations show that a nematicon is (weakly) stable. Solutions of the modulation equations are compared with numerical solutions of the full governing equations, and good agreement is found when the light intensity is small.

LIGHT SELF-LOCALIZATION IN NEMATIC LIQUID CRYSTALS: MODELLING SOLITONS IN NONLOCAL REORIENTATIONAL MEDIA

We review the mathematical modelling of propagation and specific interactions of solitary beams in nematic liquid crystals — so-called nematicons. The theory is first developed for the evolution of a single nematicon; then it is extended to the interaction of two nematicons of different wavelengths, employing linear momentum conservation equations to predict that two colour nematicons can form a vector bound state. Considering optical vortices, we show that the nonlocal response of liquid crystals stabilises a single vortex, unstable in local media. Moreover, the interaction with a nematicon in another colour can stabilise a vortex for nonlocalities far below those at which an isolated vortex remains unstable. When multiple nematicons of the same wavelength interact, the radiation they shed can join them together, still resulting in a vortex. Finally, we discuss the escape of a nematicon from a nonlinear waveguide, using simple modulation theory based on momentum conservation to model the effect and get excellent agreement with the experimental results.

Vector vortex solitons in nematic liquid crystals

We analyze the existence and stability of two-component vector solitons in nematic liquid crystals for which one of the components carries angular momentum and describes a vortex beam. We demonstrate that the nonlocal, nonlinear response can dramatically enhance the field coupling leading to the stabilization of the vortex beam when the amplitude of the second beam exceeds some threshold value. We develop a variational approach to describe this effect analytically.

Modulation solutions for the Benjamin-Ono equation

Abstract In this work, modulation solutions for three initial value problems for the Benjamin–Ono equation are studied. The first problem studied is the dispersive resolution of a step initial condition. An explicit solution of Gurevich–Pitaevskii type is derived, which explains the dispersive resolution of the step in terms of modulations. The second problem is the dispersive resolution of a breaking initial condition, while the third problem is the generation of a second phase as the result of the evolution of a modulated single phase wave. Again explicit modulation solutions for these problems are derived. In the case of the third problem, the modulation solution explicitly exhibits the formation of the new phase, in contrast to situation for the Korteweg–de Vries equation, for which the formation of a second phase has only been solved for phase loss in the dispersive analogue of shock merging. These explicit solutions are possible since the modulation theory of Dobrokhotov and Krichever for the BO equation gives uncoupled modulation equations for the different phases, which is not the case for the KdV equation. This de-coupling means that the matching of known explicit solutions enables the full description of two-phase problems. This extends results which have recently been obtained from analytic solutions of the Whitham equations for the KdV equation for shock formation and shock merging.

Synaptic Integration in Electrically Coupled Neurons

Interactions among chemical and electrical synapses regulate the patterns of electrical activity of vertebrate and invertebrate neurons. In this investigation we studied how electrical coupling influences the integration of excitatory postsynaptic potentials (EPSPs). Pairs of Retzius neurons of the leech are coupled by a nonrectifying electrical synapse by which chemically induced synaptic currents flow from one neuron to the other. Results from electrophysiology and modeling suggest that chemical synaptic inputs are located on the coupled neurites, at 7.5 microm from the electrical synapses. We also showed that the space constant of the coupled neurites was 100 microm, approximately twice their length, allowing the efficient spread of synaptic currents all along both coupled neurites. Based on this cytoarchitecture, our main finding was that the degree of electrical coupling modulates the amplitude of EPSPs in the driving neurite by regulating the leak of synaptic current to the coupled neurite, so that the amplitude of EPSPs in the driving neurite was proportional to the value of the coupling resistance. In contrast, synaptic currents arriving at the coupled neurite through the electrical synapse produced EPSPs of constant amplitude. This was because the coupling resistance value had inverse effects on the amount of current arriving and on the impedance of the neurite. We propose that by modulating the amplitude of EPSPs, electrical synapses could regulate the firing frequency of neurons.

Reaction-diffusion equations in one dimension: particular solutions and relaxation

Abstract Reaction-diffusion equations for the concentration of one species in one spatial dimension are considered, where the diffusion coefficient, as well as creation and annihilation terms are monomials. When the exponent of the annihilation term is smaller than the one of the creation term, unstable equilibrium solutions may exist. In the opposite case, travelling wave solutions are found in phase space, yielding the value of speed, even when the wave form cannot be written in closed form. The stability of some solutions is studied numerically, showing their robustness. In the case of nonlinear diffusion it is found that localized initial conditions evolve to the stable steady state via the travelling waves. The invasion of the stable state into the unstable one is via a travelling front with the largest possible gradient. This front can be interpreted as the final result of the growth of the most unstable mode. The exact solutions allow a modulation approach to study the problem with slowly varying coefficients. It is shown that when the creation term is of compact support in space, equilibria of compact support are also obtained.

Pulse evolution for a two-dimensional Sine-Gordon equation

The evolution lump and ring solutions of a Sine-Gordon equation in two-space dimensions is considered. Approximate equations governing this evolution are derived using a pulse or ring with variable parameters in an averaged Lagrangian for the Sine-Gordon equation. It was found by Neu [Physica D 43 (1990) 421] that angular variations of the pulse shape may stabilise it. However, no study of the radiation produced by the pulse was available. In the present work, the coupling of the pulse to the shed radiation is considered. It is shown both asymptotically and numerically that the angular dependence produces spiral waves which shed angular momentum, leading to the ultimate collapse of the pulse. Good quantitative agreement between the asymptotic and numerical solutions is found. In addition, it is shown how the results of the present work can be applied to the Baby Skyrme model. In this regard, it is shown how the non-zero degree of solutions of the Baby Skyrme model prevents the collapse of a non-zero degree pulse shedding zero degree radiation. It is also indicated how the present results could be applied to the study of vortex models. The analysis presented in this work shows how complicated behaviour due to radiation of angular momentum can be captured in simple terms by approximate equations for the relevant degrees of freedom.

Quantum collapse of a small dust shell

The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by Hajicek and Kuchar ˇ. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self-adjoint mo- mentum operator in the space of the shell motion which is confined to a half-line. The first problem is solved by identifying and neglecting orbits of small action in order to obtain a single valued Hamiltonian. The second problem is solved by introducing an appropriate lapse function. The resulting quantum mechanics is then studied by means of analytical and numerical techniques. We find that the region of total collapse has a very small probability. We also find that the solution concentrates around the classical Schwarzschild radius. The present work obtains from first principles a quantum mechanics for the shell and provides numerical solutions, whose behavior is explained by a detailed WKB analysis for a wide class of collapsing shells.

A discontinuous Steklov problem with an application to water waves

Abstract A discontinuous Steklov problem associated with second order selfadjoint elliptic equations which arise in water wave problems is discussed. Here it is assumed that the eigenvalue equation is satisfied only on a part of the boundary. Eigenfunction expansions are derived in an elementary way using a suitable Greens functions. Also a minimax characterization of the eigenvalues is given.