Darboux Transformation Research Articles

Modulation instability, state transitions and dynamics of multi-peak rogue wave in a higher-order coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation provides an effective description of the propagation of four optical fields in the fibers. The present work explores the dynamics of rogue waves that arise in a coupled nonlinear Schrödinger equation of higher-order that is acquired using the Ablowitz-Kaup-Newell-Segur method. The generalized (m,N−m)-fold Darboux transformation is used to determine the exact rogue wave solutions, which show multiple peaks and depressions. In addition, a detailed analysis is conducted of the modulation instability of three different kinds of plane-wave solutions. The results indicate that the coefficients of the second and third derivative terms, rather than just the coefficients of the third derivative term, have an impact on modulation instability. Lastly, the transition between rogue waves and solitons under the influence of higher-order dispersion effects is elucidated, with explicit soliton solutions provided within the modulation stability region.

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Rogue waves in a reverse space nonlocal nonlinear Schrödinger equation

Rogue waves in a reverse space nonlocal nonlinear Schrödinger (NLS) equation with real and parity-symmetric nonlinearity-induced potential are considered. This equation has clear physical meanings since it can be derived from the Manakov system with a special reduction. The N-fold Darboux transformation and its generalized form for the nonlocal NLS equation are constructed. As an application, the multiparametric Nth-order rogue wave solution in terms of Schur polynomials for the nonlocal NLS equation with focusing case is derived by the limit technique. The significant differences of rogue wave dynamics between the nonlocal NLS equation and its usual (local) counterpart are illustrated through two types of specific rogue wave solutions. Unlike the eye-shaped (Peregrine type) rogue waves, the rogue wave doublets which involve an eye-shaped rogue wave and a dark/four-petaled rogue wave merging or separating with each other, and the rogue wave sextets that are characterized by the superpositions of three eye-shaped rogue waves and three dark/four-petaled rogue waves with fundamental, triangular and quadrilateral patterns are shown. Moreover, some wave characteristics including the difference between the light intensity and the plane-wave background, and the pulse energy of the rogue wave doublets are discussed.

Singular examples of the Matrix Bochner Problem

The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem.

Generalized [formula omitted]-fold Darboux transformation and localized waves for an integrable reduced spin Hirota-Maxwell-Bloch system in an erbium doped fiber

In this paper, under investigation is an integrable reduced spin Hirota-Maxwell-Bloch (rsHMB) system, which describes the transmission of the femtosecond pulses in an erbium doped fiber. Based on the existing N-fold Darboux transformation, we establish a generalized (n,N−n)-fold Darboux transformation for the rsHMB system, where n and N are the positive integers (n≤N). The Nth-order degenerate soliton and Nth-order rogue-wave solutions for the rsHMB system are obtained via the generalized (1,N−1)-fold Darboux transformation. By means of the generalized (2,N−2)-fold Darboux transformation, we derive the Nth-order hybrid wave solutions describing the interaction between the (N−1)th-order rogue wave and the first-order breather. The above solutions are discussed graphically, which exhibit the diverse wave structures including the triangle and pentagon, among others.

Interaction of solitons in the matrix nonlinear Schrödinger equation

In this paper, we study a matrix nonlinear Schrödinger (MNLS) equation with high-order effects. Through the Darboux transformation, we construct soliton solutions of the MNLS equation up to the second order. Based on the analytic solutions, proper parameters are chosen to display evolution and elastic interaction of the solitons.

Binary Darboux transformation and N-dark solitons for the defocusing Kundu-Eckhaus equation in an optical fiber

Binary Darboux transformation and N-dark solitons for the defocusing Kundu-Eckhaus equation in an optical fiber

Solutions of the functional difference Toda equation from centered Darboux transformation

Solutions of the functional difference Toda equation from centered Darboux transformation

Multi-parametric solutions to the functional difference KdV equation

Using a specific Darboux transformation, we construct solutions to the functional difference KdV equation in terms of Casorati determinants. We give a complete description of the method and the corresponding proofs. We construct explicitly some solutions for the first orders.

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

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

Voltage manipulation of magnetic rogue waves for controllable electromagnetic interference

In this paper, we investigated the excitation of magnetic rogue waves and their voltage control in ferromagnetic nanowires. Starting from the Maxwell equations coupled with the Landau–Lifshitz–Gilbert equation, we derived the generalized derivative nonlinear Schrödinger equation under the long-wave approximation, which describes the nonlinear wave evolution in anisotropic ferromagnetic nanowires. By constructing the corresponding Lax pair, we obtained the exact solutions of first-order and second-order magnetic rogue waves via a generalized Darboux transformation. After spectral analysis and modulation instability analysis, we present the excitation regions of magnetic rogue waves. Furthermore, we tested and achieved precise control over the occurrence positions of first-order and second-order magnetic rogue waves by applying bias voltage. We further discussed the physical implications of these results, highlighting their potential applications in controlled electromagnetic interference for data destruction.

Soliton, breather and rogue wave solutions of the higher-order modified Gerdjikov–Ivanov equation

In this paper, we investigate the dynamics of localized waves for the higher-order modified Gerdjikov–Ivanov equation. Based on Lax pair, the Nth-fold Darboux transformation is constructed. The soliton, breather and rogue wave solutions are obtained and presented graphically. Moreover, the dynamic properties of the above localized waves are analyzed. Specially, the interactions between solitons and bound states are achieved. In addition, the rational W-shaped soliton is derived by the degeneration of breather solutions.

Darboux transformation-based LPNN generating novel localized wave solutions

Darboux transformation method is one of the most essential and important methods for solving localized wave solutions of integrable systems. In this work, we introduce the core idea of Darboux transformation of integrable systems into the Lax pairs informed neural networks (LPNNs), which we proposed earlier. By fully utilizing the data-driven solutions, spectral parameter and spectral function obtained from LPNNs, we present the novel Darboux transformation-based LPNN (DT-LPNN). The notable feature of DT-LPNN lies in its ability to solve data-driven localized wave solutions and spectral problems with high precision, and it also can employ Darboux transformation formulas of integrable systems and non-trivial seed solutions to discover novel localized wave solutions that were previously unobserved and unreported. The numerical results indicate that, by utilizing the single-soliton solutions as the non-trivial seed solutions, we obtain novel localized wave solutions for the Kraenkel–Manna–Merle (KMM) system by employing DT-LPNN, in which solution u changes from original bright single-soliton on zero background wave to new dark single-soliton dynamic behavior on a variable non-zero background wave. Moreover, by treating the two-soliton solutions as the non-trivial seed solutions, DT-LPNN generates novel localized wave solutions for the KMM system that exhibit completely different dynamic behaviors from prior two-soliton solutions. DT-LPNN combines the Darboux transformation theory of integrable systems with deep neural networks, offering a new approach for generating novel localized wave solutions using non-trivial seed solutions.

Soliton and breather solutions of a reverse time nonlocal coupled nonlinear Schrödinger equation with four-wave mixing effect

Under investigation is a reverse time nonlocal coupled nonlinear Schrödinger equation with four-wave mixing effect. The N-fold Darboux transformation is constructed in a compact determinant representation. As an application, the multi-bright–bright soliton, multi-dark–bright soliton and multi-breather solutions are presented. Dynamical behaviors of the degenerate bright–bright solitons, the regular and periodically oscillating dark–bright solitons and the multiple breather interactions are graphically shown.

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

This Letter 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.

Soliton, breather and rogue wave solutions of the nonlinear Schrödinger equation via Darboux transformation on a time–space scale

Solving soliton equations on the time–space scale has always been a challenging issue. In this paper, we firstly generalize the Ablowitz–Kaup–Newel–Segur (AKNS) method to the time–space scale, concurrently obtain the nonlinear Schrödinger (NLS) equation on this scale, which unifies the continuous and the semi-discrete NLS equations. On this basis, the N-fold Darboux transformation is proposed for the NLS equation on a space scale. As applications, soliton, breather, and rogue wave solutions of NLS equation are derived from diverse seed solutions on a space scale. Specially, the rouge solution on a space scale is obtained for the first time.

The optical solitons for the three-component Dirac–Manakov system via the Darboux transformation

The optical solitons for the three-component Dirac–Manakov system via the Darboux transformation

Influences of damping, perturbation and variable coefficient on an extended nonlinear Gardner model

Describing the nonlinear wave propagation in fluids has posed a challenging task. The nonlinear Gardner equation stands as a crucial model in fluid dynamics. An extended Gardner model incorporates variable coefficients, damping, and perturbation terms, offering a more comprehensive approach to understanding complex wave dynamics in fluids. In our study, we employ the symbolic computation tool, specifically the Darboux transformation, to derive new analytic solutions up to the Nth order for this extended Gardner equation. By assigning the spectrum parameters in these solutions into real values, we are able to obtain line-like solitons. Moreover, when N⩾2 and pairs of complex conjugate values are assigned to two of the spectrum parameters, breathers emerge. This enables a comprehensive analysis of the dynamics involving solitons, breathers, and their hybrid forms, shedding light on their interactions. Furthermore, our work introduces several noteworthy insights by investigating the influences of three key factors: the damping, perturbation, and variable coefficient. Through a few of specific examples, we can confirm the following facts: (i) The damping factor plays a crucial role in rendering solitons and breathers unstable, causing them to undergo exponential changes in both amplitudes and directions during propagation. (ii) The perturbation function creates non-zero background waves along the time axis, which interact with solitons and breathers. (iii) The variable coefficient can introduce disturbances to solitons and breathers. Our findings disclose that this equation exhibits a wealth of intricate propagation patterns. Various factors can exert varying degrees of influence on solitons and their dynamics. This observation just underscores the complexity inherent in the real world and should be useful in applications.

Darboux transformations for Dunkl–Schrödinger equations with energy-dependent potential and position-dependent mass

We construct arbitrary-order Darboux transformations for Schrödinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates our equations with the standard Schrödinger case. We apply our method to generate several solvable Dunkl–Schrödinger equations.

Darboux transformation of symmetric Jacobi matrices and Toda lattices

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J(p) = UL (or J(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.