Kaup–Newell Research Articles

The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.

Higher-order mixed solution and breather solution on a periodic background for the Kundu equation

In this paper, we use the generalized Darboux transformation (GDT) method and the Taylor expansion to construct solutions for the Kundu equation including the multiple breather solution on a periodic background and the mixed solutions of multiple soliton and multiple breather solution. Starting from the Lax pair under the framework of Kaup–Newell (KN) system, we obtain an odd-fold Darboux transformation. Then by means of the Taylor expansion, the odd-fold GDT is obtained in the form of determinants. By selecting different parameters in the determinants, we find various solutions. Modulation instability (MI) for the Kundu equation is also studied as a small aspect.

Higher dimensional reciprocal integrable Kaup-Newell systems

The study of integrable systems is one of important topics both in physics and in mathematics. However, traditional studies on integrable systems are usually restricted in (1+1) and (2+1) dimensions. The main reasons come from the fact that high-dimensional integrable systems are extremely rare. Recently, we found that a large number of high dimensional integrable systems can be derived from low dimensional ones by means of a deformation algorithm. In this paper, the (1+1) dimensional Kaup-Newell (KN) system is extended to a (4+1) dimensional system with the help of the deformation algorithm. In addition to the original (1+1) dimensional KN system, the new system also contains three reciprocal forms of the (1+1) dimensional KN system. The model also contains a large number of new (<i>D</i>+1) dimensional (<inline-formula><tex-math id="M2">\begin{document}$D \leqslant 3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222418_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222418_M2.png"/></alternatives></inline-formula>) integrable systems. The Lax integrability and symmetry integrability of the (4+1) dimensional KN system are also proved. It is very difficult to solve the new high-dimensional KN systems. In this paper, we only investigate the traveling wave solutions of a (2+1) dimensional reciprocal derivative nonlinear Schrödinger equation. The general envelope travelling wave can be expressed by a complicated elliptic integral. The single envelope dark (gray) soliton of the derivative nonlinear Schödinger equation can be implicitly written.

Integrable couplings of two expanded non-isospectral soliton hierarchies and their bi-Hamiltonian structures

Starting from two matrix spectral problems associated with the real special orthogonal Lie algebra [Formula: see text], two non-isospectral hierarchies of Ablowitz–Kaup–Newell–Segur (AKNS) type and Kaup–Newell (KN) type are constructed. In addition, we enlarge the two spectral problems and then obtain non-isospectral [Formula: see text] integrable couplings of AKNS type and KN type by solving the expanded non-isospectral zero curvature equation. We find that the obtained four hierarchies have the bi-Hamiltonian structure of the combined form. It follows that all equations in the resulting soliton hierarchy are integrable in the sense of Liouville.

The generalized super Kaup–Newell equation and its super bi-Hamiltonian structure

In this work, from the generalized Kaup–Newell (KN) equation spectral problem, the generalized super KN equation and its bi-Hamiltonian structure are constructed based on a Lie super-algebras and the super-trace identity. Besides, a generalized super KN equation with self-consistent sources is also discussed.

A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup–Newell equation

Due to the fact that the higher-order Kaup–Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann–Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single–double zeros, and double–double zeros are especially shown.

Several exact solutions of the reduced fourth-order flow equation of the Kaup–Newell system

In this paper, for the Kaup–Newell (KN) system, by making use of the determinant representations of n-fold Darboux transformation (DT) and n-transformed solutions, we obtain similar conclusions for the fourth-order flow equation of the KN system (FOFEKN). Furthermore, on one hand, starting from the zero seed solution q=0, we obtain several exact solutions of the reduced FOFEKN, such as soliton solutions and positon solutions. On the other hand, if we start with the periodic non-zero seed solution, then we obtain some breather solutions and rogue wave solutions of the reduced FOFEKN.

Bright-dark and multi wave novel solitons structures of Kaup-Newell Schrödinger equations and their applications

In this paper, the soliton solutions which indicate long wave parallel to the magnetic fields of Kaup-Newell models are argued via described method. Modified extended Simple equation is suggested to explore the novel solitons and other wave structures of two different types of Kaup-Nawell equations, which have never been constructed before. As a consequence, bright-dark solitons, singular solitons, multi-wave solitons, breather type wave of strange structures and other waves solutions of two Kaup-Newell (K-N) Schrödinger equations are achieved in different form. The obtained novel solitons and other exact wave solutions have key applications in engineering and applied physics. Novel wave structures of solitons are explained graphically by providing suitable values to parameters that help for understanding the physical phenomena of these models. The constructed solutions are evaluated with available results in the literature. This technique can be productively utilized to more equations that occur in mathematical physics.

Optical solitons and other solutions to Kaup–Newell equation with Jacobi elliptic function expansion method

In this paper, we studied Kaup–Newell (KN) equation in coupled vector form without four-wave mixing terms in birefringent fibers. We employed Jacobi elliptic function expansion method in order to demonstrate sub-pico-second optical soliton solutions. Beside bright and dark solitons, Jacobi elliptic function solutions and hyperbolic solutions are obtained. Moreover, the graphs for some solution are presented.

Several categories of exact solutions of the third-order flow equation of the Kaup–Newell system

In this paper, we introduce the third-order flow equation of the Kaup–Newell (KN) system. We study this equation, and we obtain different types of solutions by using the Darboux transformation (DT) and the extended DT of the KN system, such as solitons, positons, breathers, and rogue waves. The extended DT is obtained by taking the degenerate eigenvalues $$ \lambda _{i} \rightarrow \lambda _{1} (i=3,5,7,\ldots ,2k-1)$$ and by performing the Taylor expansion near $$\lambda _{1}$$ of the determinants of DT. Some analytic expressions are explicitly given for the first-order solutions. We study the unique waveforms of both the first-order and higher-order rogue-wave solutions for special choices of parameters, and we find different types of such wave structures: fundamental pattern, triangular, modified-triangular, pentagram, ring, ring-triangular, and multi-ring wave patterns. We conclude that the third-order dispersion and quintic nonlinear term of the KN system modify both the trajectories and speeds of the solutions as compared with those corresponding to the second-order flow equation of the KN system.

Exact solutions of nonlocal Fokas–Lenells equation

In this paper we propose a nonlocal Fokas–Lenells (FL) equation which can be derived from the Kaup–Newell (KN) linear scattering problem. By constructing the Darboux transformation of nonlocal FL equation, we obtain its different kinds of exact solutions including bright/dark solitons, kink solutions, periodic solutions and several types of mixed soliton solutions. It is shown that the solutions of nonlocal FL equation possess different properties from the normal FL equation.

The τ-symmetry algebra of the generalized Kaup–Newell hierarchy

Abstract In this paper, based on the generalized Kaup–Newell(KN) spectral problem, the corresponding isospectral and nonisospectral soliton hierarchies are derived. It is shown that its Lie algebraic structure of the corresponding zero curvature equation is worked out. Finally the τ − symmetry algebra of the isospectral and nonisospectral generalized KN hierarchies is deduced.

Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup–Newell lattice equation

Under consideration in this paper is the Kaup–Newell (KN) lattice equation which is an integrable discretization of the KN equation. Infinitely, many conservation laws and discrete N-fold Darboux transformation (DT) for this system are constructed and established based on its Lax representation. Via the resulting N-fold DT, the discrete multi-dark soliton solutions in terms of determinants are derived from non-vanishing background. Propagation and elastic interaction structures of such solitons are shown graphically. Overtaking interaction phenomena between/among the two, three and four solitons are discussed. Numerical simulations are used to explore their dynamical behaviors of such multi-dark solitons. Numerical results show that their evolutions are stable against a small noise. Results in this paper might be helpful for understanding the propagation of nonlinear Alfvén waves in plasmas.

Modulational instability and dynamics of implicit higher-order rogue wave solutions for the Kundu equation

Under investigation in this paper is the Kundu equation, which may be used to describe the propagation process of ultrashort optical pulses in nonlinear optics. The modulational instability of the plane-wave for the possible reason of the formation of the rogue wave (RW) is studied for the system. Based on our proposed generalized perturbation [Formula: see text]-fold Darboux transformation (DT), some new higher-order implicit RW solutions in terms of determinants are obtained by means of the generalized perturbation [Formula: see text]-fold DT, when choosing different special parameters, these results will reduce to the RW solutions of the Kaup–Newell (KN) equation, Chen–Lee–Liu (CLL) equation and Gerjikov–Ivanov (GI) equation, respectively. The relevant wave structures are shown graphically, which display abundant interesting wave structures. The dynamical behaviors and propagation stability of the first-order and second-order RW solutions are discussed by using numerical simulations, the higher-order nonlinear terms for the Kundu equation have an impact on the propagation instability of the RW. The method can also be extended to find the higher-order RW or rational solutions of other integrable nonlinear equations.

On integrable wave interactions and Lax pairs on symmetric spaces

Multi-component generalizations of derivative nonlinear Schrödinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann–Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup–Newell (KN) and Gerdjikov–Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end.

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations**Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2012AQ011, ZR2013AL016, ZR2015EM042, National Social Science Foundation of China under Grant No. 13BJY026, the Development of Science and Technology Project under Grant No. 2015NS1048 and A Project

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, including the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.

Nonlinear Integrable Couplings of the Kaup-Newell Hierarchy

In this paper, a six-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which an isospectral problem is established. By zero curvature equations, we obtain the nonlinear integrable couplings of the Kaup-Newell (KN) hierarchy.

Nonlinear Integrable Couplings of the Kaup-Newell Hierarchy

In this paper, a six-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which an isospectral problem is established. By zero curvature equations, we obtain the nonlinear integrable couplings of the Kaup-Newell (KN) hierarchy.

The algebraic structure behind the derivative nonlinear Schrödinger equation

The Kaup–Newell (KN) hierarchy contains the derivative nonlinear Schrödinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of an Kac–Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows. The equivalence between the latter and the massive Thirring model is also explicitly demonstrated. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.

Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics

Conservation law plays a vital role in the study of nonlinear evolution equations, particularly with regard to integrability, linearization and constants of motion. In the present paper, it is shown that infinitely many conservation laws for certain nonlinear evolution equations are systematically constructed with symbolic computation in a simple way from the Riccati form of the Lax pair. Note that the Lax pairs investigated here are associated with different linear systems, including the generalized Kaup–Newell (KN) spectral problem, the generalized Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem, the generalized AKNS–KN spectral problem and a recently proposed integrable system. Therefore, the power and efficiency of this systematic method is well understood, and we expect it may be useful for other nonlinear evolution models, even higher-order and variable-coefficient ones.