Binary Darboux Transformation Research Articles

Binary Darboux transformation of vector nonlocal reverse-space nonlinear Schrödinger equations

For vector nonlocal reverse-space nonlinear Schrödinger equations, a binary Darboux transformation is formulated by using two sets of eigenfunctions and adjoint eigenfunctions. The resulting binary Darboux transformation has been decomposed into an [Formula: see text]-fold product of single binary Darboux transformations. An application starting from zero seed potentials generates a class of soliton solutions.

The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation. We treat real or complex-valued functions, such as g 1 = g 1(x, t) and g 2 = g 2(x, t) as non-commutative, and employ the Lax pair associated with the evolution equation, as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation. The soliton solutions are presented explicitly within the framework of quasideterminants. To visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles. Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.

Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $\tau$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Optical fiber communication plays a crucial role in modern communication. In this work, we focus on the higher-order matrix nonlinear Schrödinger equation with negative coherent coupling in a birefringent fiber. For the slowly varying envelopes of two interacting optical modes, we construct a binary Darboux transformation using the corresponding Lax pair. With vanishing seed solutions and the binary Darboux transformation, we investigate vector degenerate soliton and exponential soliton solutions. By utilizing these soliton solutions, we demonstrate three types of degenerate solitons and double-hump bright solitons. Furthermore, considering non-vanishing seed solutions and applying the binary Darboux transformation, we obtain vector breather solutions, and present the vector single-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, double-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, and vector degenerate beak-type breathers. Additionally, we take the limit in the breather solutions and derive vector rogue wave solutions. We illustrate the beak-type rogue waves and bright-dark rogue waves. Humps of these vector double-hump waves can separate into two individual humps. The results obtained in this work may potentially provide valuable insights for experimentally manipulating the separation of two-hump solitons, breathers, and rogue waves in optical fibers.

A binary Darboux transformation for multi-component nonlinear Schrödinger equations and dark vector soliton solutions

By taking the plane wave potentials as the seed solutions, we harness a binary Darboux transformation to generate dark vector soliton solutions for multi-component nonlinear Schrödinger equations. We introduce a generalized Darboux matrix such that the eigenvalues could equal the adjoint eigenvalues. The method which is purely algebraic could be useful and convenient, particularly in the construction of dark soliton solutions of integrable systems.

Quasi-Grammian solutions of the coupled Gerdjikov–Ivanov equation

We construct the N-fold standard binary Darboux transformation (bDT) for the coupled Gerdjikov–Ivanov (cGI) equation. Then, using the bDT presented in this article, we explicitly construct the exact solutions of the cGI equation in terms of quasi-Grammians. As applications of the bDT, we present various particular solutions for the cGI equation, including soliton, breather, and rogue wave solutions.

Binary Darboux transformation and interactions of solitons for a higher-order matrix nonlinear Schrödinger equation

In this paper, we present two- and three-soliton solutions for a higher-order matrix nonlinear Schrödinger equation based on the binary Darboux transform constructed, and analyze various types of interaction properties of solitons. For the two solitons, our analytical results reveal four major features that are associated to interaction dynamics: breather-like solitons, bound-state solitons, bound states of the solitons with one/two peaks and bound states of one soliton and another soliton with breather-like structure. In addition, we graphically analyze the related properties with regard to interactions of three solitons, and seven different types of interesting appearances about soliton interactions are obtained, including shape-changing and shape-preserving interactions.

Solitons and quasi-Grammians of the generalized lattice Heisenberg magnet model

In this paper, we study the discrete Darboux and standard binary Darboux transformation for the generalized lattice Heisenberg magnet model. We calculate the quasi-Grammian solutions by the iteration of standard binary Darboux transformation. Furthermore, we derive the explicit matrix solutions for the binary Darboux matrix and then reduce them to the elementary Darboux matrix and plot the dynamics of solutions.

Binary Darboux transformation, solitons and breathers for a second-order three-wave resonant interaction system

Binary Darboux transformation, solitons and breathers for a second-order three-wave resonant interaction system

Vector localized and periodic waves for the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation

Dynamical properties of vector localized and periodic waves hold significant importance in the study of physical systems. In this work, we investigate the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation. For the two interacting components, we construct the binary Darboux transformation formulas, vector localized, and periodic wave solutions. Via those solutions, different kinds of nonlinear waves can be achieved, including rogue waves, solitons, positons, and periodic waves. When the imaginary part of the spectral parameter is not zero, eye-shaped rogue waves appear in one component, and the twisted rogue wave pairs in the other component. As the spectral parameter is real, we derive distinct forms of vector localized and periodic waves on the non-zero background, such as the vector solitons, positons, periodic waves, breathers on the periodic wave background, and rational solitons. These results may be valuable in this investigation of nonlinear waves in physical systems.

A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation

AbstractIn the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step‐type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.

Dynamics of the rogue lump in the asymmetric Nizhnik–Novikov–Veselov system

AbstractThe (2 + 1)‐dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation describes an incompressible fluid such as waves with weakly nonlinear restoring forces, long internal waves in a density‐stratified ocean, or acoustic waves on a crystal lattice. In this paper, we construct a new kind of solutions doubly localized in both time and space, that is, a rogue lump wave, of the ANNV equation using the binary Darboux transformation (BDT). This solution is expressed by a class of semirational functions, in which lumps appear from one line soliton and then rapidly disappear into another line soliton. Phase parameter newly introduced in eigenfunctions plays a vital role in controlling the distinct dynamical behaviors of these rogue lumps. The explicit formulas of the approximate locations and heights of rogue waves, lumps, and line solitons are given by asymptotic analysis. Their collision process is discussed in detail. These results further help us to understand the extreme and transient waves in a variety of physical systems.

A vectorial binary Darboux transformation for the first member of the negative part of the AKNS hierarchy

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member of the ‘negative’ part of the AKNS hierarchy. A reduction leads to the first ‘negative flow’ of the NLS hierarchy, which in turn is a reduction of a rather simple nonlinear complex PDE in two dimensions, with a leading mixed third derivative. This PDE may be regarded as describing geometric dynamics of a complex scalar field in one dimension, since it is invariant under coordinate transformations in one of the two independent variables. We exploit the correspondingly reduced vectorial binary Darboux transformation to generate multi-soliton solutions of the PDE, also with additional rational dependence on the independent variables, and on a plane wave background. This includes rogue waves.

Relativistic perfect fluids: integrability of the classical Taub system

It is established that the classical Taub system governing isentropic planar motions of perfect fluids in special relativity may be formulated entirely in soliton-theoretic terms. The well-established tools of integrable systems theory may therefore be used to produce large classes of solutions given in parametric form. As an illustration, the general solution of the Taub system associated with a privileged pressure-density constitutive law is generated by applying a binary Darboux transformation to the seed solution of the Taub system descriptive of extreme fluids.

A Comprehensive Study of the Complex mKdV Equation through the Singular Manifold Method

In this paper, we introduce a modification of the Singular Manifold Method in order to derive the associated spectral problem for a generalization of the complex version of the modified Korteweg–de Vries equation. This modification yields the right Lax pair and allows us to implement binary Darboux transformations, which can be used to construct an iterative method to obtain exact solutions.

Two-component complex modified Korteweg–de Vries equations: New soliton solutions from novel binary Darboux transformation

Two-component complex modified Korteweg–de Vries equations: New soliton solutions from novel binary Darboux transformation

Interaction dynamics of optical dark bound solitons for a defocusing Lakshmanan-Porsezian-Daniel equation.

We investigate the propagation and interaction dynamics of the optical dark bound solitons for the defocusing Lakshmanan-Porsezian-Daniel equation, which is a physically relevant generalization of the nonlinear Schrödinger equation involving the higher-order effects. Explicit N-dark soliton solutions in the compact determinant form are constructed via the binary Darboux transformation method. Bound states of the dark solitons are discussed when the incoherent solitons have the same velocity. We find an interesting phenomenon that dark soliton molecules and double-valley dark solitons (DVDSs) can be obtained by controlling the interval of the bound state dark solitons, and abundant interaction modalities between them can be formed. Moreover, dark soliton molecules always undergo elastic interactions with other solitons, while interactions for the DVDSs are usually inelastic, and special parameter conditions for elastic interaction of DVDSs through asymptotic analysis are obtained. Numerical simulations are employed to verify the stability of the bound state dark solitons. Analytical results obtained in this paper are expected to be useful for the experimental realization of bound-state dark solitons in optical fibers with higher-order effects and a further understanding of their optical transmission properties..

Darboux transformation and interaction solutions of localized waves for a defocusing coupled complex modified Korteweg-de Vries equation

In this paper, the binary Darboux transformation for a defocusing coupled complex modified Korteweg-de Vries (dccmKdV) equation was investigated. With nonzero plane wave initial conditions, we further derived its dark–dark solitons, breather solutions, different types of rogue waves and some new interaction structures among them. These novel results can be applied in the areas of fluid, plasma mechanics, etc.

Binary Darboux transformation, solitons, periodic waves and modulation instability for a nonlocal Lakshmanan–Porsezian–Daniel equation

In this paper, a nonlocal Lakshmanan–Porsezian–Daniel equation is investigated with the help of the binary Darboux transformation method and asymptotic analysis. Nonlocality of that equation has been reflected in that the solutions of that equation at the location ς depend on both the local solution at ς and the nonlocal solution at −ς, where ς is the retarded time coordinate. We derive the formulas of the Nth-order solutions through the obtained binary Darboux transformation, where N is a positive integer. Under certain conditions, the first-order periodic waves and solitons are obtained, e.g., degenerate solitons, dark–dark solitons, bright–bright solitons and dark–bright solitons. Interactions between/among the dark solitons, bright solitons and periodic wave are discussed and graphically illustrated. We discuss the modulation instability of that equation.

Binary Darboux transformation and new soliton solutions of the focusing nonlocal nonlinear Schrödinger equation

For the focusing nonlocal nonlinear Schrödinger (NNLS) equation, we establish the N-fold binary Darboux transformation (BDT) and give a complete proof on the form invariance of Lax pair. Then, by choosing the tanh-function solution as a seed, we implement the one-fold BDT and obtain two new types of soliton solutions including the exponential and exponential-and-rational types. Both two types of solutions are nonsingular with a wide range of parameter regimes, and they can describe the elastic soliton interactions over the nonzero background with the asymptotic phase difference π as x→±∞. Also, we derive the expressions of all asymptotic solitons and discuss the parametric conditions for different soliton profiles. It turns out that each asymptotic soliton can display both the dark and antidark soliton profiles or exhibit no spatial localization, which confirms that the focusing NNLS equation admits a rich variety of soliton interactions like the defocusing NNLS case.