International Journal of Modern Physics B

A survey is made which highlights recent topics on the dynamics of algebraic solitons, which are exact solutions to a certain class of nonlinear integrodifferential evolution equations. The model equations that we consider here are the Benjamin-Ono (BO) and its higher-order equations together with the BO-Burgers equation, a model equation for deep-water waves, the sine-Hilbert (sH) equation and a damped sH equation. While these equations have their origin either in physics or in mathematics, each equation exhibits a novel type of algebraic soliton solution and hence its characteristic is worth studying in its own right. After deriving these equations, we are concerned with each equation separately. We first present explicit N-soliton solutions and then summarize related mathematical properties of the equation. Subsequently, a detailed description is given to the interaction process of two algebraic solitons using the pole expansion of the solution. Particular attention is paid to investigating the effects of small perturbations on the overtaking collision of two BO solitons by employing a direct multisoliton perturbation theory. It is shown that the dynamics of interacting algebraic solitons reveal new aspects which have never been observed in the interaction process of usual solitons expressed in terms of exponential functions.

We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.

In Douady and Couder’s experiment [Phys. Rev. Lett. 68, 2098, (1992)], spiral phyllotactic-like patterns were formed by magnetic dipoles dropped sequentially into a vertical magnetic field. The dipoles move radially with constant velocity limited by frictional forces in a silicone oil. This paper presents an analysis of the stability of this self-organized growth process using a second nearest neighbor (SNN) interaction approximation. A recurrence formula is written for the divergence angle; i.e. the angle defined by successive radial vectors. Further analysis of the dynamics reveals conditions for convergence of the recurrence formula. The limit of the divergence angle sequence includes a fractional part of 2π which is the ratio of pairs of Fibonacci numbers. It also includes a term which varies linearly with a growth parameter µ, defined in the text. The system is stable within the SNN approximation. Even richer dynamical behaviors, including that observed experimentally, require higher order approximations.

Resonant tunneling is studied theoretically in asymmetrical double-barrier structures with arbitrary potential profile. Analytical expressions of the transmission coefficient and resonance condition are derived by taking into account the mass difference between the well and barrier layers. It is confirmed that resonant tunneling with unity resonant transmission or under-unity resonant transmission may occur in asymmetrical double-barrier structures. Two independent conditions are required for unity resonant transmission: one is the Phase Difference Condition for Resonance (PDCR) and the other is the Maximum Condition for the Peak Value (MCPV). The under-unity resonant transmission occurs when only PDCR holds. Furthermore, wave functions of an electron at resonance level are calculated and the confining phenomenon is studied.