Generalized Darboux Transformation Research Articles

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrodinger equation

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrodinger equation

Predicting positon solutions of a family of nonlinear Schrödinger equations through deep learning algorithm

We consider a hierarchy of nonlinear Schrödinger equations (NLSEs) and forecast the evolution of positon solutions using a deep learning approach called Physics Informed Neural Networks (PINN). Notably, the PINN algorithm accurately predicts positon solutions not only in the standard NLSE but also in other higher order versions, including cubic, quartic and quintic NLSEs. The PINN approach also effectively handles two coupled NLSEs and two coupled Hirota equations. In addition to the above, we report exact second-order positon solutions of the sextic NLSE and coupled generalized NLSE. These solutions are not available in the existing literature and we construct them through generalized Darboux transformation method. Further, we utilize PINNs to forecast their behaviour as well. To validate PINN's accuracy, we compare the predicted solutions with exact solutions obtained from analytical methods. The results show high fidelity and low mean squared error in the predictions generated by our PINN model.

On a Hirota equation in oceanic fluid mechanics: Double-pole breather-to-soliton transitions

In oceanic fluid mechanics, a Hirota equation describing the propagation of the deep ocean broader-banded waves is studied in this paper. With respect to the envelope of the wave field, we simultaneously take the multi-pole phenomena and breather-to-soliton transitions into account to investigate the double-pole breather-to-soliton transitions via the second-order generalized Darboux transformation. Under certain parameter conditions, double-pole anti-dark solitons, periodic waves, W-shaped solitons and multi-peak solitons are derived, analyzed and graphically illustrated. We perform the asymptotic analysis on the double-pole anti-dark solitons to investigate their interaction properties, e.g., amplitudes, characteristic lines, slopes, phase shifts and position differences. Different from the known double-pole solitons of some other models, the double-pole anti-dark solitons, hereby, indicate that the two soliton components own unequal amplitudes.

Double-pole dark-bright mixed solitons for a three-wave-resonant-interaction system

This paper explores a three-wave-resonant-interaction system that is widely seen in fluid mechanics, plasma physics and nonlinear optics. With the aid of the iterated-form generalized Darboux transformation method, we obtain the second-order analytic solutions that illustrate the double-pole dark-bright mixed solitons. Under certain parameter conditions, the double-pole bright-dark-bright solitons are illustrated through the 3D, density and contour plots. Those plots demonstrate the stable attractive forces between the adjacent soliton components. Moreover, we conduct the asymptotic analysis on the double-pole bright-dark-bright solitons to investigate their physical properties, which are consistent with those in the plots. Our work might contribute to the studies of the dynamic properties of the curved dark-bright solitons and be extended to the double-pole dark-bright mixed solitons within the other integrable systems.

Taking into consideration a fifth-order nonlinear Schrödinger equation in an optical fiber

In this paper, symbolic computation on a fifth-order nonlinear Schrödinger equation is done, for the attosecond pulses propagation in an optical fiber. With respect to the complex amplitude of the optical pulse envelope, we work out a Lax pair and derive the modified generalized Darboux transformation. Then, we give the semirational solutions via the modified generalized Darboux transformation method. By means of such solutions, we graphically discuss the properties for three types of the degenerate solitons.

Large-time lump patterns of Kadomtsev-Petviashvili I equation in a plasma analyzed via vector one-constraint method

In plasma physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for investigating the evolution characteristics of nonlinear waves. For the KPI equation, the constraint method is an effective tool for generating solitonic or rational solutions from the solutions of lower-dimensional integrable systems. In this work, various nonsingular, rational lump solutions of the KPI equation are constructed by employing the vector one-constraint method and the generalized Darboux transformation of the (1 + 1)-dimensional vector Ablowitz–Kaup–Newell–Segur system. Furthermore, we investigate the large-time asymptotic behavior of high-order lumps in detail and discover distinct types of patterns. These lump patterns correspond to the high-order rogue wave patterns of the (1 + 1)-dimensional vector integrable equation and are associated with root structures of generalized Wronskian–Hermite polynomials.

Multi-breathers and higher-order rogue waves on the periodic background in a fourth-order integrable nonlinear Schrödinger equation

In this paper, we present a systematic formulation of multi-breathers and higher-order rogue wave solutions of a fourth-order nonlinear Schrödinger equation on the periodic background. First of all, we compute a complete family of elliptic solution of this higher-order equation, which can degenerate into two particular cases, i.e., the dnoidal and cnoidal solutions. By using the modified squared wavefunction approach, we solve the spectral problem on the elliptic function background. Then, we derive multi-breather solutions in terms of the theta functions, particular examples of which are the Kuznetsov-Ma breather and the Akhmediev breather. Furthermore, taking the limit of the breather solutions at branch points, we construct higher-order rogue wave solutions by employing a generalized Darboux transformation technique. On the periodic background, we present the first-order, second-order and second-second-order rogue waves. With aid of the theta functions, we explicitly characterize the resulting breathers and rogue waves, and demonstrate their dynamic behaviors by illustrative examples. Finally, we discuss how the parameter of the higher-order effects affects the breathers and rogue waves.

Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.

Modulation instability and localized wave excitations for a higher-order modified self-steepening nonlinear Schrödinger equation in nonlinear optics

This paper investigates a higher-order modified nonlinear Schrödinger equation with higher-order dispersion and self-steepening effects, which can be used to study the dynamics of asymmetric and steepened optical pulse transmission in optical fibres. The modulation instability of the plane-wave solution has been analysed, and the state transition of the rogue waves under the higher-order dispersion effect is presented. The findings demonstrate that the self-steepening effect may also produce an asymmetric unstable frequency band. Combined with the modulation instability, the rogue wave, rational soliton and mixed interaction of localized waves have a quantitative relationship with plane-wave parameters. Various exact solutions can be accurately located and obtained according to the generalized Darboux transformation. The asymptotic analysis of rational solutions demonstrates the state transition of higher-order rogue waves. This paper illustrates the significance of modulation instability for studying the integrable systems by the Darboux transformation. It is a new guidance to solve the difficulty of exact excitation of asymmetric localized waves in complex integrable systems. The results have prospective applications and references for the emergence, amplification and compression of asymmetric pulses in optical experiments.

Rational solitons and rogue waves for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated.

Nth order generalized Darboux transformation and solitons, breathers and rogue waves in a variable-coefficient coupled nonlinear Schrödinger equation

Nth order generalized Darboux transformation and solitons, breathers and rogue waves in a variable-coefficient coupled nonlinear Schrödinger equation

N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics

N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics

Localized wave solutions to the vector nonlinear Schrödinger equation with nonzero backgrounds

The vector nonlinear Schrödinger (NLS) system is studied through the generalized Darboux transformation (DT). We begin with the nonzero seed solution in the exponential form of evolution variable and explore the eigenfunction of the ‐component NLS equation. With the formulae of the eigenfunction, localized wave solutions of the two‐component and three‐component NLS equations are studied. By choosing different free parameters, some special solutions and the dynamic evolutions are presented, including breather–rogue wave, breather fission, breather fusion, and breather–soliton.

On a variable-coefficient AB system in a baroclinic flow: Generalized Darboux transformation and non-autonomous localized waves

In this paper, we focus our attention on a variable-coefficient AB system, which models the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we construct an N-fold generalized Darboux transformation (GDT) via symbolic computation, where N is a positive integer. Via the obtained GDT, several kinds of the non-autonomous localized waves, e.g., the multi-pole solitons, multi-pole breathers, rogue waves and their interactions, are investigated. We have selected four types of the variable coefficients, i.e., constant, linear about t, quadratic about t and trigonometric about t, where t is the normalized retarded time coordinate. We explore the influence of those selected variable coefficients on the non-autonomous localized waves. Moreover, we discuss the bound states among the non-autonomous multiple-pole solitons and one soliton, which exhibit the periodic attractions and repulsions between the adjacent solitons.

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

Inhomogeneous reduced Maxwell-Bloch system in nonlinear optics: Darboux-transformation and solitonic issues

Studies on nonlinear optics are active, providing the applications in modulators, fiber lasers, optical sensors, etc. In this paper, we investigate an inhomogeneous reduced Maxwell-Bloch system, which could describe the ultra-short optical pulses through an inhomogeneous medium. N-fold Darboux transformation is constructed via an existing Lax pair, where N is a positive integer. Generalized Darboux transformation is obtained via a limit procedure. One- and two-soliton solutions are determined with the aid of the N-fold Darboux transformation. Double-pole soliton solutions are derived via the generalized Darboux transformation. With respect to the electric field, microscopic polarization, phase information, and atomic inversion related to the inhomogeneous medium, one soliton, interaction between the two solitons, and double-pole soliton are graphically depicted. Effects of the coefficients in that system on those solitons are shown via the trigonometric and linear functions.

Hybrid structures of localized waves for the coupled extended mKdV equation

For the coupled extended modified Korteweg–de Vries (mKdV) equation, we investigate hybrid structure waves composed of localized waves. This equation can be considered as the generalization of the mKdV equation. Firstly, hybrid structures of localized waves can be derived in view of the generalized Darboux transformation. Next, we get different interactions between those hybrid waves with the parameters changing. After that, it is found that under a certain condition, the hybrid solutions will degenerate into rational solutions. In particular, we find an interesting rational rogue wave with the novel dynamical behavior. In addition, we show the striking dynamics of hybrid solutions by analyzing each figure. These new solutions can help us investigate integrable systems better.

Generalized Darboux algorithms for Dunkl–Schrödinger equations with position-dependent mass at fixed energy

We apply generalized Darboux transformations to Schrödinger equations at fixed energy that contain Dunkl derivatives, and are equipped with a position-dependent mass. Our Darboux transformations, originating from coupled Korteweg–De Vries systems, consist of two algorithms that generate new solutions, potentials and position-dependent mass functions for the Dunkl–Schrödinger system.

Generalized Darboux transformation and rogue waves for a coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber

Generalized Darboux transformation and rogue waves for a coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber

Degenerate and bound-state solitons of a novel Kundu-nonlinear Schrödinger equation based on generalized Darboux transformation

In this paper, we derive a novel Kundu-nonlinear Schrödinger equation through a 2 × 2 matrix spectral problem, which can be used to describe the phenomena in optical propagation, and the determinant expression of generalized (n,N−n)-fold Darboux transformation for the Kundu-nonlinear Schrödinger equation is given by solving algebraic equations. In addition, through the generalized (n,N−n)-fold Darboux transformation, we obtain three different types of solutions, (i) soliton solutions, (ii) degenerate soliton solutions, (iii) interaction solutions between solitons and degenerate solitons, which are divided into elastic interaction and bound-state. At the same time, the conditions for their generation are discussed. The properties of the aforementioned solutions are analyzed graphically ultimately.