Prolongation theory. A new nonlinear Schrödinger equation

We discuss a new kind of nonlinear Schrödinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation.     

Fluctuating solitons of the KdV equations

We introduce a new AKNS three-component system, which is convenient for finding periodic and/or almost periodic solutions to the hierarchy of the KdV equations. It conserves the spectral functions which determine the spectrum of the auxiliary Schrödinger equation containing the solutions of the Korteweg-de Vries equations as potentials. By means of the Darboux and Abraham-Moses transformations we derive new solutions of the KdV hierarchy, which can be grasped as solitons on the fluctuating background.Some parts of the paper were delivered in the talk at the III Potsdam-V Kiev international workshop on nonlinear processes in physics, Potsdam (USA), 1–11 August, 1991.     

Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation

We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton.     

On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach

We give a Lie superalgebraic interpretation of the biHamiltonian structure of known supersymmetric KdV equations. We show that the loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the supersymmetric KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra.     

Exact Solutions for Self-Dual SU(2) and SU(3) Yang–Mills Fields

The (constrained) canonical reduction of four-dimensional self-dual SU(2) and SU(3) Yang–Mills theory to two-dimensional nonlinear Schrödinger (NS) and Korteweg–de Vries (KdV) equations are considered. The Bäcklund transformations (BTs) are implemented to obtain new classes of exact solutions for the reduced two-dimensional NS and KdV models.     

An alternative derivation of the pulse solutions of the KdV hierarchy

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.     

Super-Matrix KdV and Super-Generalized NS Equations from Self-Dual Yang–Mills Systems with Supergauge Groups

Super-matrix KdV and super-generalized nonlinear Schrödinger equations are shown to arise from a symmetry reduction of ordinary self-dual Yang–Mills equations with supergauge groups.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Higher-order integrable evolution equation and its soliton solutions

We consider an extended nonlinear Schrödinger equation with higher-order odd and even terms with independent variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying the Darboux transformation, present its first and second order soliton solutions. The equation and its solutions have two free parameters. Setting one of these parameters to zero admits two limiting cases: the Hirota equation on the one hand and the Lakshmanan–Porsezian–Daniel (LPD) equation on the other hand. When both parameters are zero, the limit is the nonlinear Schrödinger equation.     

Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation

We investigated the soliton solution for N$N$ coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management.     

A hierarchy of non-isospectral multi-component AKNS equations and its integrable couplings

A hierarchy of non-isospectral multi-component AKNS equations is derived from an arbitrary order matrix spectral problem. As a reduction, non-isospectral multi-component Schrödinger equations are obtained. Moreover, new non-isospectral integrable couplings of the resulting AKNS soliton hierarchy are constructed by enlarging the associated matrix spectral problem.     

On the continuous limit of integrable lattices I. The Kac-Moerbeke system and KdV theory

KdV theory is constructed systematically through the continuous limit of the Kac-Moerbeke system. The infinitely many commuting vector fields, the conserved functionals, the Lax pairs and the biHamiltonian structure are recovered as the limits of suitably defined linear combinations of homologous objects for the Kac-Moerbeke system. The combinatorial aspects of this recombination method are treated in detail.     

Loop Group-Valued CS and WZNW models,integrable systems,and self-dual Yang-Mills theory

Affine CS and WZNW theories with values in infinite-dimensional (loop) groups are proposed. It appears that the affine CS theory naturally introduces a spectral parameter into a CS theory. The Sinh-Gordon, KdV, and nonlinear Schrödinger equations are obtained, via Hamiltonian reductions, from the affine WZNW. It is shown that the self-dual Yang-Mills (SDYM) equation is related to the equation of motion of the affine WZNW and, thus, symmetry algebra underlying the SDYM can be identified with the affine two-loop Kac-Moody algebra of the affine WZNW.K. C. Wong Research Award Winner, address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

具有色散系数的(2+1)维非线性Schrödinger方程的有理解和空间孤子

非线性Schrödinger方程是物理学中具有广泛应用的非线性模型之一. 本文采用相似变换, 将具有色散系数的(2+1)维非线性Schrödinger方程简化成熟知的Schrödinger方程, 进而得到原方程的有理解和一些空间孤子.     

Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method

This paper presents the coupled version of a previous work on nonlinear Schrödinger equation [23]. It focuses on the construction of approximate solutions of nonlinear Schrödinger equations. In this paper, we applied the differential transformation method (DTM) to solving coupled Schrödinger equations. The obtained results show that the technique suggested here is accurate and easy to apply.     

Nonlinear wave mechanics,information theory,and thermodynamics

A logarithmic nonlinear term is introduced in the Schrödinger wave equation, and a physical justification and interpretation are provided within the context of information theory and thermodynamics. From the resulting nonlinear Schrödinger equation for a system at absolute temperatureT>0, the energy equivalence,kT 1n 2, of a bit of information is derived.     

Shape invariance and the exactness of the quantum Hamilton-Jacobi formalism

Quantum Hamilton-Jacobi theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schrödinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this Letter, we show that shape invariance also suffices to determine the eigenvalues in quantum Hamilton-Jacobi theory.     

Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formulation, and its connection with the functional Schrödinger representation in the temporal gauge are discussed. The mass gap problem is related to the finite-dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator which represents the DW Hamiltonian operator.     

Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equationcan be generated by its recursion operator.We generalize the result to a special form of Lax pair,from which a method to constrain the integrable system to alower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depictedby a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.