The Quantum Auxiliary Linear Problem & Darboux-Backlund Transformations

This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing method and Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory.The second part contains original contributions that represent new developments in the theory and applications of these transformations. Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.

Bäcklund transformation (BT) is an intrinsic property of nonlinear integrable system. Generally, it is an interesting and challenging work to investigate BT of a nonlinear system, especially for the non-polynomial form. In this paper, we introduce an analytic method for constructing BTs of the generalized nonlinear wave equations (NLWEs) of the form utt=auxx+h(u). The BTs with arbitrary parameters are provided explicitly, so the integrability of the equation is verified accordingly. Then, the nonlinear superposition formulas (NSFs) of the NLWEs are given in terms of such BTs, and the infinite number of exact explicit solutions to the equations are obtained based on the BTs and the NSFs. Furthermore, the BTs of the other types of NLWEs of the form uxt=h(u) can be provided directly by the variable transformation method.

We show the time evolution of minimum-uncertainty coherent-state (MUCS) wave packets in the solvable potentials we have considered. (Numerical techniques are discussed in the appendices.) The time evolution is compared to the motion that a classical particle would have in the same potential. We make a number of observations on the conditions which can cause the states to lose their coherence more (or less) rapidly with time, and compare the MUCS results with those which can be obtained from other types of coherent states. The most physically interesting comparison is with the continuous representation coherent states.

An analytical solution of the Schrödinger equation with open boundary conditions in all scattering channels has been found for asymmetric triple-barrier resonance-tunneling structures with thin high barriers. This solution describes resonance transitions between three quantum levels in a high rf electric field. It is found that, under certain conditions, most electrons incident on the upper resonance level can emit two photons and leave a structure through the lower level without intermediate interaction with phonons. The structure appears to be almost absolutely transparent in a wide range of the rf field amplitude. This behavior fundamentally distinguishes the multiphoton scattering process from previously considered single-photon scattering processes and the quantum efficiency of such transitions can be twice as high as the maximum quantum efficiency of the transitions between neighboring levels and can reach a value of 160% in the limiting case.

The correlation of ultracold neutral plasmas extracted from the pair correlation function is presented in detail. To get its spatial distribution and time evolution, a large scale simulation using a full molecular-dynamics method with open boundary condition is performed, and the expansion of an ultracold plasma is simulated. We demonstrate that the coupling strength extracted from pair correlation is a good measure for the correlation in ultracold plasma except at the very early stage. The ion–ion pair correlation arises at the timescale of . And results imply that there is a strong coupling area at the outer plasma during the expansion.

In view of the fact that the road capacity drops obviously near the lane reduction bottleneck, we propose two traffic cellular automata models around the merging point under two different traffic management measures. One is the control scheme of first-arrive and first-in, the other is the periodic control scheme. Using the open boundary condition, the numerical simulations are performed. The flux, density distribution and the evolution time from congested state to free state under different entering probability, deceleration zone length and periods are obtained. The comparisons indicate that whether the traffic management measures are chosen properly or not is closely related to entering probability, deceleration zone length and period. These conclusions have a significant reference value for the reduction of traffic jams near the bottlenecks.

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O( sqrt{N} ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

We study inhomogeneous quantum quenches in the attractive regime of the sine--Gordon model. In our protocol, the system is prepared in an inhomogeneous initial state in finite volume by coupling the topological charge density operator to a Gaussian external field. After switching off the external field, the subsequent time evolution is governed by the homogeneous sine--Gordon Hamiltonian. Varying either the interaction strength of the sine--Gordon model or the amplitude of the external source field, an interesting transition is observed in the expectation value of the soliton density. This affects both the initial profile of the density and its time evolution and can be summarised as a steep transition between behaviours reminiscent of the Klein--Gordon, and the free massive Dirac fermion theory with initial external fields of high enough magnitude. The transition in the initial state is also displayed by the classical sine--Gordon theory and hence can be understood by semi-classical considerations in terms of the presence of small amplitude field configurations and the appearance of soliton excitations, which are naturally associated with bosonic and fermionic excitations on the quantum level, respectively. Features of the quantum dynamics are also consistent with this correspondence and comparing them to the classical evolution of the density profile reveals that quantum effects become markedly pronounced during the time evolution. These results suggest a crossover between the dominance of bosonic and fermionic degrees of freedom whose precise identification in terms of the fundamental particle excitations can be rather non-trivial. Nevertheless, their interplay is expected to influence the sine--Gordon dynamics in arbitrary inhomogeneous settings.

In this paper we investigate the time evolution of a general driven time-dependent oscillator using the evolution operator method developed by Cheng et al. We obtain an exact form of the time evolution operator which, in turn, enables us to find the exact wave functions and coherent states at any time t. Our analyses indicate that the time-dependent coherent state is equivalent to the well-known squeezed state, while the time-dependent number state is equivalent to the displaced and squeezed number state. Besides, we also calculate the time-dependent transition probabilities among the coherent states and number states of a simple harmonic oscillator associated with the initial Hamiltonian H(0).

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

Conservation laws, heirarchies, scattering theory and Backlund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrodinger equation, modified KdV, and the n-wave equation). We first find a simple model Poisson group action that contains flows for systems with a Lax pair whose terms all decay on R. Backlund transformations and flows arise from subgroups of this single Poisson group. For the ZS-AKNS nxn heirarchy defined by a constant a ∈ u(n), the simple model is no longer correct. The a determines two separate Poisson structures. The flows come from the Poisson action of the centralizer Ha of a in the dual Poisson group (due to the behavior of e at infinity). When a has distinct eigenvalues, Ha is abelian and it acts symplectically. The phase space of these flows is the space Sa of left cosets of the centralizer of a in D−, where D− is a certain loop group. The group D− contains both a Poisson subgroup corresponding to the continuous scattering data, and a rational loop group corresponding to the discrete scattering data. The Ha-action is the right dressing action on Sa. Backlund transformations arise from the action of the simple rational loops on Sa by right multiplication. Various geometric equations arise from appropriate choice of a and restrictions of the phase space and flows. In particuar, we discuss applications to the sine-Gordon equation, harmonic maps, Schrodinger flows on symmetric spaces, Darboux orthogonal coordinates, and isometric immersions of one space-form in another. 1 Research supported in part by NSF Grant DMS 9626130 2 Research supported in part by Sid Richardson Regents’ Chair Funds, University of Texas system

Time evolution of large classical systems.- Ergodic properties of infinite systems.- Time evolution and ergodic properties of harmonic systems.- The laser: A reversible quantum dynamical system with irreversible classical macroscopic motion.- What does it mean for a mechanical system to be isomorphic to the Bernoulli flow?.- The Geodesic flow on surfaces of negative curvature.- Lectures on the billiard.- Spectral invariants and smooth ergodic theory.- Nonlinear wave equations.- Integrable systems of nonlinear evolution equations.- Discrete and periodic illustrations of some aspects of the inverse method.- Finitely many mass points on the line under the influence of an exponential potential -- an integrable system.- On traveling wave solutions of nonlinear diffusion equations.- The existence of heteroclinic orbits, and applications.- Hadamard's generalization of hyperbolicity, with applications to the hopf bifurcation problem.- Hyperbolic sets and shift automorhpisms.- Triple collision in Newtonian gravitational systems.- Solutions of the collinear four body problem which become unbounded in finite time.- On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus.

We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus–focus equilibrium. From the singularity in the actions we deduce that the Arnol'd determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parameterize the Liouville tori by their frequencies. If one perturbs this integrable system, then the KAM tori form a Whitney smooth family: they can be smoothly interpolated by a torus bundle that is diffeomorphic to the bundle of Liouville tori of the unperturbed integrable system. As is well known, this bundle of Liouville tori is not trivial. Our result implies that the KAM tori have monodromy. In semi-classical quantum mechanics, quantization rules select sequences of KAM tori that correspond to quantum levels. Hence a global labelling of quantum levels by two quantum numbers is not possible.

Generally, open string boundary conditions play a nontrivial role in string theory. For example, in the presence of an antisymmetric tensor background field, they will lead the spacetime coordinates noncommutative. In this paper, we mainly discuss how to build up a generalized Dirichlet normal ordered product of open bosonic string embedding operators that satisfies both the equations of motion and the generalized Dirichlet boundary conditions at the quantum level in the presence of an antisymmetric background field, as the generalized Neumann case has already been discussed in the literature. Further, we also give a brief check of the consistency of the theory under the newly introduced normal ordering.

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