We propose a simple algebraic method for generating classes of traveling wave solutions for a variety of partial differential equations of current interest in nonlinear science. This procedure applies equally well to equations which may or may not be
integrable. We illustrate the method with two distinct classes of models, one with solutions including compactons in a class of models inspired by the Rosenau-Hyman, Rosenau-Pikovsky and Rosenau-Hyman-Staley equations, and the other with solutions including peakons in a system which generalizes the Camassa-Holm, Degasperis-Procesi and Dullin-Gotwald-Holm equations. In both cases, we obtain new classes of solutions not studied before.
The structure of supersymmetry is analyzed systematically in ${cal PT}$ symmetric quantum mechanical theories. We give a detailed description of supersymmetric systems associated with one dimensional ${cal PT}$ symmetric quantum mechanical theories.
We show that there is a richer structure present in these theories compared to the conventional theories associated with Hermitian Hamiltonians. We bring out various properties associated with these supersymmetric systems and generalize such quantum mechanical theories to higher dimensions as well as to the case of one dimensional shape invariant potentials.
We present a detailed analysis of our recent observation that the origin of the geometric tachyon, which arises when a D$p$-brane propagates in the vicinity of a stack of coincident NS5-branes, is due to the proper acceleration generated by the backg
round dilaton field. We show that when a fundamental string (F-string), described by the Nambu-Goto action, is moving in the background of a stack of coincident D$p$-branes, the geometric tachyon mode can also appear since the overall conformal mode of the induced metric for the string can act as a source for proper acceleration. We also studied the detailed dynamics of the F-string as well as the instability by mapping the Nambu-Goto action of the F-string to the tachyon effective action of the non-BPS D-string. We qualitatively argue that the condensation of the geometric tachyon is responsible for the (F,D$p$) bound state formation.
General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in
physical systems with potentials involving spin and non-local interactions.
We study the question of stability of the ground state of a scalar theory which is a generalization of the phi^3 theory and has some similarity to gravity with a cosmological constant. We show that the ground state of the theory at zero temperature b
ecomes unstable above a certain critical temperature, which is evaluated in closed form at high temperature.
The motion of a Dp-brane in the background of a stack of coincident NS5-branes is analysed as the motion of a relativistic point particle in the transverse space of the five-branes. In this system, the particle experiences a proper acceleration ortho
gonal to its proper velocity due to the background dilaton field which changes the dynamics from that of a simple geodesic motion. In particular, we show that in the vicinity of the five-branes, it is this acceleration which is responsible for modifying the motion of the radial mode to that of an inverted simple harmonic oscillator leading to the tachyonic instability.
We study the question of diagonalizability of the Hamiltonian for the Faddeev-Reshetikhin (FR) model in the two particle sector. Although the two particle S-matrix element for the FR model, which may be relevant for the quantization of strings on $Ad
S_{5}times S^{5}$, has been calculated recently using field theoretic methods, we find that the Hamiltonian for the system in this sector is not diagonalizable. We trace the difficulty to the fact that the interaction term in the Hamiltonian violating Lorentz invariance leads to discontinuity conditions (matching conditions) that cannot be satisfied. We determine the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral Gross-Neveu model which we find share the same S-matrix. We explain this by showing, through a Fierz transformation, that these two models are in fact equivalent. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle S-matrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is $PT$ symmetric. We show that the wave functions for this system are also $PT$ symmetric. Thus, the theory is in a $PT$ unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix.
Combining the thermal operator representation with the dispersion relation in QED at finite temperature and chemical potential, we determine the complete retarded photon self-energy only from its absorptive part at zero temperature. As an application
of this method, we show that, even for the case of a nonzero chemical potential, the temperature dependent part of the one loop retarded photon self-energy vanishes in $(1+1)$ dimensional massless QED.
