Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.

Non-invasive measurement of trigeminal nerve somatosensory evoked potentials

Whiplash injuries are common yet enigmatic to substantiate clinically. Trigeminal somatosensory evoked potentials (TSEPs) were posited as an indicator of trigeminal nerve conduction damage resulting from whiplash. Alternating polarity square-wave current stimuli were applied transcutaneously in the facial region. 379 recorded pilot trials from 27 participants (8 male and 19 female) were utilized to develop a non-invasive recording capability for TSEPs. Stimulus intensity and artifact, cortical recording sites, stimulation electrode design and placement were explored. Statistically significant differences in amplitude of TSEP waveform components at 13, 19 and 27 ms between uninjured and whiplashed participants were noted. Increased stimulus intensity in whiplashed participants was observed to increase TSEP amplitude. The present methodology and hardware are discussed and directions for future advancement of the current process are outlined.

Some Travelling Wave Solutions of KdV-Burgers Equation

In this paper we study the extended Tanh method to obtain some exact solutions of KdV-Burgers equation. The principle of the Tanh method has been explained and then apply to the nonlinear KdV- Burgers evolution equation. A finnite power series in tanh is considered as an ansatz and the symbolic computational system is used to obtain solution of that nonlinear evolution equation. The obtained solutions are all travelling wave solutions.