AKNS Research Articles

The system of the vortex-like structures: The viewpoint on a turbulence modeling

In this study we suggest new approach to turbulence modeling. To develop this approach, we construct the set of the vortex-like dynamical systems evolving in the space E3. These systems are constructed using the AKNS hierarchy so that they generalize thin closed vortex filament described in the local induction approximation. We construct and investigate the Hamiltonian dynamics of these systems in terms of the non-standard variables. We formulate the principles of quantization these systems focusing on the quantum description of the filaments with linearized dynamics. To describe the energy of vortex-like structures, we apply a new method based on the use of the central extended Galilei group. In a special case, we calculate the energy levels of a system of quantized vortices. The proposed approach makes it possible to describe both the interaction of such vortex-like filaments and the processes of their creation and annihilation in a natural way. As a result, we have constructed a quantum turbulent flow model that demonstrates the relationship between stochastic and integrable dynamics.

Multiple mixed solutions of the nonlocal sine-Gordon equation

In this paper, we propose the nonlocal sine-Gordon equation from the AKNS system by the Parity and the Time symmetries. Moreover, the main work is to construct the Darboux transformation for the nonlocal sine-Gordon equation. In particular, various types of solutions are obtained by taking a seed solution, such as, soliton solutions, kink solutions, mixed solutions.

Integrable Coupling of Expanded Isospectral and Non-Isospectral Dirac Hierarchy and Its Reduction

In this paper, we first generalize the Dirac spectral problem to isospectral and non-isospectral problems and use the Tu scheme to derive the hierarchy of some new soliton evolution equations. Then, integrable coupling is obtained by solving the isospectral and non-isospectral zero curvature equations.We find that the obtained hierarchy has the bi-Hamiltonian structure of the combined form. In particular, one of the integrable soliton hierarchies is reduced to be similar to the coupled nonlinear Schördinger system in the AKNS hierarchy. Next, the strict self-adjointness of the reduced equation system is verified, and conservation laws are constructed with the aid of the Ibragimov method. In addition, we apply the extended Kudryashov method to obtain some exact solutions of this reduced equation system.

On a coupled nonlocal nonlinear Schrödinger system

Recently, Ablowitz and Musslimani have introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable coupled nonlocal nonlinear Schrödinger equation which can be derived from the AKNS system. The Darboux transformation is constructed for this equation. Via this Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, periodic solutions and so on. Dynamics and interactions of different kinds of soliton solutions are discussed. Finally, we compare the obtained results with standard coupled NLS equation.

Single-Soliton Solution of KdV Equation via Hirota’s Direct Method under the Time Scale Framework

Hirota’s direct method is one significant way to obtain solutions of soliton equations, but it is rarely studied under the time scale framework. In this paper, the generalized KdV equation on time-space scale is deduced from one newly constructed Lax equation and zero curvature equation by using the AKNS method, which can be reduced to the classical and discrete KdV equation by considering different time scales. What is more, it is the first time that the single-soliton solution of the KdV equation under the time scale framework is obtained by using the idea of Hirota’s direct method.

Lax Operator Algebras and Applications to τ-Symmetries for Multilayer Integrable Couplings

The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As illustrative examples, isospectral and non-isospectral integrable couplings and the corresponding commutator relations of their Lax operators are computed explicitly in the cases of the Volterra lattice hierarchy and the AKNS hierarchy, along with their τ-symmetry algebras.

Elliptic solitons and “freak waves”

It is shown that elliptic solutions to the AKNS hierarchy equations can be obtained by exploring spectral curves that correspond to elliptic solutions of the KdV hierarchy. This also allows one to get the quasirational and trigonometric solutions for AKNS hierarchy equations as a limit case of the elliptic solutions mentioned above.

Riemann–Hilbert problems of a nonlocal reverse-time AKNS system of six-order and dynamical behaviors of N-soliton

In this paper, we solve nonlinear nonlocal reverse-time six-component sixth-order AKNS system. We used reverse-time reduction to reduce the coupled system to an integrable sixth-order nonlinear Schrödinger-type equation. Starting from the spectral problem of the AKNS system, a Riemann–Hilbert problem will be formulated. This formulation allows us to generate soliton solutions by using the vectors lying in the kernel of the matrix Jost solutions. When reflection coefficients are zeros, the jump matrix is identity and the corresponding Riemann–Hilbert problem yields soliton solutions, leading to explore their dynamics.

Darboux Transformations and Soliton-Like Solutions of a New System Associated with the Negative AKNS System

In this letter, we consider a new two-component system. By its reciprocal relation with the first negative flow in the AKNS hierarchy, we construct three Darboux-Bäcklund transformations for the new system and obtain some soliton-like solutions.

Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

One-phase elliptic solutions of the nonlocal nonlinear equations from AKNS hierarchy and their spectral curves

After publishing the pioneering works of Ablowitz and Musslimani, other authors also began active research on nonlocal forms of classical integrable nonlinear equations. They usually investigate particular equations, and for these equations they construct solutions that are expressed in terms of elementary functions. In present paper, we investigate one-phase elliptic solutions of all the equations from the AKNS hierarchy, including mixed ones. We also analyze the properties of spectral curves of the considered one-phase solutions in order to be able to construct multiphase algebro-geometric solutions of nonlocal forms of the AKNS hierarchy equations in the future.

Superposition of the coupled NLS and MKdV systems

Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same N-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in (1+1)-dimensions.

Affine Lie Superalgebra and Lax Equations Associated with B(0, 1)(1)

By using the Lie superalgebra B(0, n ), we give a Cartan matrix of affine Lie superalgebra B(0, n)(1), and the relations of their generators. We also give an example of super AKNS hierarchy associated with B(0, n). As an application of affine Lie superalgebra B(0, n)(1), we obtain two kinds of Lie-superalgebraic interpretations of super AKNS hierarchy. We also get conservation laws associated with super AKNS hierarchy.

Tire Tracks and Integrable Curve Evolution

Abstract We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation—the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the Ablowitz, Kaup, Newell, and Segur (AKNS) system and the vortex filament equation. We show that “bicycle correspondence” of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a three-dimensional bike of imaginary length. We show that a series of examples of “ambiguous” closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

AbstractIn this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg‐de Vries hierarchy, and nonlocal versions of the sine‐Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two‐soliton solutions of the reverse‐t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.

Generalized Matrix Exponential Solutions to the AKNS Hierarchy

Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure.

An integrable hierarchy including the AKNS hierarchy and its strict version

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl2 into Lie subalgebras g≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl2 and represent it as C = C≥0⊕C<0. We deform the Lie subalgebras C≥0 and C<0 by the respective groups corresponding to g<0 and g≥0. Further, we require that the evolution equations of the deformed generators of C≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.

AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients

We apply the AKNS hierarchy to derive the generalized KdV equation andthe generalized modified KdV equation with variable-coefficients. We system-atically derive new exact solutions for them. The solutions turn out to beexpressible in terms of doubly-periodic Jacobian elliptic functions.

The Strict AKNS Hierarchy: Its Structure and Solutions

We discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space ofsl2than that in the AKNS case and whose Lax equations are based on a different decomposition of this loop space. We show the compatibility of these Lax equations and that they are equivalent to a set of zero curvature relations. We present a linearization of the system and conclude by giving a wide construction of solutions of this hierarchy.

A hierarchy of super AKNS hierarchy related to Lie superalgebra sl(2|1) and a finite dimensional super Hamiltonian system

A hierarchy of super AKNS equations associated with a [Formula: see text] matrix-valued spectral problem is derived. It is shown that each equation in the hierarchy is bi-super Hamiltonian. Moreover, a new finite dimensional super Hamiltonian system (FDSHS), together with its Lax representation, [Formula: see text]-matrix and conversed integrals of motion, is obtained from the spectral problem by binary nonlinearization.