Solitary Wave Interactions In Dispersive Equations Using Manton's Approach

We generalize the approach first proposed by Manton [Nuc. Phys. B {\bf 150}, 397 (1979)] to compute solitary wave interactions in translationally invariant, dispersive equations that support such localized solutions. The approach is illustrated using as examples solitons in the Korteweg-de Vries equation, standing waves in the nonlinear Schr{\"o}dinger equation and kinks as well as breathers of the sine-Gordon equation.Comment: 5 pages, 4 figures, slightly modified version to appear in Phys. Rev.

Restoring Vision through “Project Prakash”: The Opportunities for Merging Science and Service

“So how does this help society?” is a question we are often asked as scientists. The lack of immediate and tangible results cannot be held against a scientific project but statements of future promise in broad and inchoate terms can sometimes pass the benefit-buck indefinitely. There is no incentive against over-stating the benefits, especially when they are hypothetical and lie in the distant future. Few scientists will say their science is not designed to serve society. Yet the proliferation of “potential benefits” in grant proposals and the Discussion sections of research papers, in the absence of tangible translations, can make the service element of science seem like a cliched ritual. Its repetition hollows out its meaning, breeding cynicism about the idea that basic science can be of service

Comment on "Exact results for survival probability in the multistate Landau-Zener model"

We correct the proof of Brundobler-Elser formula (BEF) provided in [2004 \textit{J. Phys. B: At. Mol. Opt. Phys.} \textbf{37} 4069] and continued in Appendix of [2005 \textit{J. Phys. B: At. Mol. Opt. Phys.} \textbf{38} 907]. After showing that some changes of variables employed in these articles are used erroneously, we propose an alternative change of variables which solves the problem. In our proof, we reveal the connection between the BEF for a general $N$-level Landau-Zener system and the exactly solvable bow-tie model. The special importance of the diabatic levels with maximum/minimum slope is emphasized throughout.Comment: 10 page

Beyond the KdV: post-explosion development

Several threads of the last 25 years’ developments in nonlinear wave theory that stem from the classical Korteweg–de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a nonlocal integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors’ view of the future development of the chosen lines of nonlinear wave theory

Landau level broadening in graphene with long-range disorder -- Robustness of the n=0 level

Broadening of the Landau levels in graphene and the associated quantum Hall plateau-to-plateau transition are investigated numerically. For correlated bond disorder, the graphene-specific n=0 Landau level of the Dirac fermions becomes anomalously sharp accompanied by the Hall transition exhibiting a fixed-point-like criticality. Similarly anomalous behavior for the n=0 Landau level is also shown to occur in correlated random magnetic fields, which suggests that the anomaly is generic to disorders that preserve the chiral symmetry.Comment: 4 pages, 5 figures, submitted to EP2DS-18 Conference proceeding

Localized wave structures: Solitons and beyond

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and “sublinear” nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton–soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles (“soliton gas”), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves

Counterintuitive transitions in the multistate Landau-Zener problem with linear level crossings

We generalize the Brundobler-Elser hypothesis in the multistate Landau-Zener problem to the case when instead of a state with the highest slope of the diabatic energy level there is a band of states with an arbitrary number of parallel levels having the same slope. We argue that the probabilities of counterintuitive transitions among such states are exactly zero.Comment: 9 pages, 5 figure

Thermal diffusion of solitons on anharmonic chains with long-range coupling

We extend our studies of thermal diffusion of non-topological solitons to anharmonic FPU-type chains with additional long-range couplings. The observed superdiffusive behavior in the case of nearest neighbor interaction (NNI) turns out to be the dominating mechanism for the soliton diffusion on chains with long-range interactions (LRI). Using a collective variable technique in the framework of a variational analysis for the continuum approximation of the chain, we derive a set of stochastic integro-differential equations for the collective variables (CV) soliton position and the inverse soliton width. This set can be reduced to a statistically equivalent set of Langevin-type equations for the CV, which shares the same Fokker-Planck equation. The solution of the Langevin set and the Langevin dynamics simulations of the discrete system agree well and demonstrate that the variance of the soliton increases stronger than linearly with time (superdiffusion). This result for the soliton diffusion on anharmonic chains with long-range interactions reinforces the conjecture that superdiffusion is a generic feature of non-topological solitons.Comment: 11 figure

Subcritical multiplicative chaos for regularized counting statistics from random matrix theory

For an N×N random unitary matrix U_N, we consider the random field defined by counting the number of eigenvalues of U_N in a mesoscopic arc of the unit circle, regularized at an N-dependent scale Ɛ_N>0. We prove that the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. In addition, we show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. The proofs are based on the asymptotic analysis of certain Toeplitz or Fredholm determinants using the Borodin-Okounkov formula or a Riemann-Hilbert problem for integrable operators. Our approach to the L¹-phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context