Negaton and Positon solutions of the soliton equation with self-consistent sources

The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for $N$-times repeated GBDT are presented. This GBDT provides non-auto-B\"{a}cklund transformation between two KdV equations with different degrees of sources and enable us to construct more general solutions with $N$ arbitrary $t$-dependent functions. By taking the special $t$-function, we obtain multisoliton, multipositon, multinegaton, multisoliton-positon, multinegaton-positon and multisoliton-negaton solutions of KdVES. Some properties of these solutions are discussed.Comment: 13 pages, Latex, no figues, to be published in J. Phys. A: Math. Ge

Generalized Darboux transformations for the KP equation with self-consistent sources

The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time $t$ for the KPESCS which, in contrast with the binary Darboux transformation of the KP equation, provides a non-auto-B\"{a}cklund transformation between two KPESCSs with different degrees. The formula for N-times repeated generalized binary Darboux transformation is proposed and enables us to find the N-soliton solution and lump solution as well as some other solutions of the KPESCS.Comment: 20 pages, no figure

The Solutions of the NLS Equations with Self-Consistent Sources

We construct the generalized Darboux transformation with arbitrary functions in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is proposed. By reduction the generalized Darboux transformation with arbitrary functions in time $t$ for the Nonlinear Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and enables us to find the dark soliton, bright soliton and positon solutions for NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.Comment: 24 pages, 3 figures, to appear in Journal of Physics A: Mathematical and Genera

Conductance of 1D quantum wires with anomalous electron-wavefunction localization

We study the statistics of the conductance $g$ through one-dimensional disordered systems where electron wavefunctions decay spatially as $|\psi| \sim \exp (-\lambda r^{\alpha})$ for $0 <\alpha <1$, $\lambda$ being a constant. In contrast to the conventional Anderson localization where $|\psi| \sim \exp (-\lambda r)$ and the conductance statistics is determined by a single parameter: the mean free path, here we show that when the wave function is anomalously localized ($\alpha <1$) the full statistics of the conductance is determined by the average $$ and the power $\alpha$. Our theoretical predictions are verified numerically by using a random hopping tight-binding model at zero energy, where due to the presence of chiral symmetry in the lattice there exists anomalous localization; this case corresponds to the particular value $\alpha =1/2$. To test our theory for other values of $\alpha$, we introduce a statistical model for the random hopping in the tight binding Hamiltonian.Comment: 6 pages, 8 figures. Few changes in the presentation and references updated. Published in PRB, Phys. Rev. B 85, 235450 (2012

Anderson localization of one-dimensional hybrid particles

We solve the Anderson localization problem on a two-leg ladder by the Fokker-Planck equation approach. The solution is exact in the weak disorder limit at a fixed inter-chain coupling. The study is motivated by progress in investigating the hybrid particles such as cavity polaritons. This application corresponds to parametrically different intra-chain hopping integrals (a "fast" chain coupled to a "slow" chain). We show that the canonical Dorokhov-Mello-Pereyra-Kumar (DMPK) equation is insufficient for this problem. Indeed, the angular variables describing the eigenvectors of the transmission matrix enter into an extended DMPK equation in a non-trivial way, being entangled with the two transmission eigenvalues. This extended DMPK equation is solved analytically and the two Lyapunov exponents are obtained as functions of the parameters of the disordered ladder. The main result of the paper is that near the resonance energy, where the dispersion curves of the two decoupled and disorder-free chains intersect, the localization properties of the ladder are dominated by those of the slow chain. Away from the resonance they are dominated by the fast chain: a local excitation on the slow chain may travel a distance of the order of the localization length of the fast chain.Comment: 31 pages, 13 figure

Giant oscillations of energy levels in mesoscopic superconductors

The interplay of geometrical and Andreev quantization in mesoscopic superconductors leads to giant mesoscopic oscillations of energy levels as functions of the Fermi momentum and/or sample size. Quantization rules are formulated for closed quasiparticle trajectories in the presence of normal scattering at the sample boundaries. Two generic examples of mesoscopic systems are studied: (i) one dimensional Andreev states in a quantum box, (ii) a single vortex in a mesoscopic cylinder.Comment: 4 pages, 3 figure

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let \Gamma be an arithmetic lattice in G and let C=C(\Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \hat{F}_{\omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example \Gamma=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the ring of S-integers in k, with S=\{v\}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of \Gamma on the Bruhat-Tits tree associated with G.Comment: 27 pages, 5 figures, to appear in J. Reine Angew. Mat

Enhanced Transmission Through Disordered Potential Barrier

Effect of weak disorder on tunneling through a potential barrier is studied analytically. A diagrammatic approach based on the specific behavior of subbarrier wave functions is developed. The problem is shown to be equivalent to that of tunneling through rectangular barriers with Gaussian distributed heights. The distribution function for the transmission coefficient $T$ is derived, and statistical moments \left are calculated. The surprising result is that in average disorder increases both tunneling conductance and resistance.Comment: 10 pages, REVTeX 3.0, 2 figures available upon reques

Integrable dispersionless KdV hierarchy with sources

An integrable dispersionless KdV hierarchy with sources (dKdVHWS) is derived. Lax pair equations and bi-Hamiltonian formulation for dKdVHWS are formulated. Hodograph solution for the dispersionless KdV equation with sources (dKdVWS) is obtained via hodograph transformation. Furthermore, the dispersionless Gelfand-Dickey hierarchy with sources (dGDHWS) is presented.Comment: 15 pages, to be published in J. Phys. A: Math. Ge