Darboux Transformation Research Articles

Scaling symmetry reductions of coupled KdV systems

Abstract In this paper we discuss the Painlevé reductions of coupled KdV systems. We start by comparing the procedure with that of stationary reductions. Indeed, we see that exactly the same construction can be used at each step and parallel results obtained. For simplicity, we restrict attention to the t2 flow of the KdV and DWW hierarchies and derive respectively 2 and 3 compatible Poisson brackets, which have identical structure to those of their stationary counterparts. In the KdV case, we derive a discrete version, which is a non-autonomous generalisation of the well known Darboux transformation of the stationary case.

Integrability and Darboux Transformation of a Four-component Volterra Lattice Equation

Integrability and Darboux Transformation of a Four-component Volterra Lattice Equation

An extended (2+1)-dimensional modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation: Lax pair and Darboux transformation

Abstract The aim of this paper is to study an extended modified Korteweg-de Vries Calogero-Bogoyavlenskii-Schiff (mKdV-CBS) equation and present its Lax pair with a spectral parameter. Meanwhile, a Miura transformation is explored, which reveals the relationship between solutions of the extended mKdV-CBS equation and the extended (2+1)-dimensional Korteweg-de Vries (KdV) equation. On the basis of the obtained Lax pair and the existing research results, Darboux transformation is derived, which plays a crucial role in presenting soliton solutions. Besides, soliton molecules are given by the velocity resonance mechanism.

Dark solitons on elliptic function background for the defocusing Hirota equation

Abstract We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms. By means of the modified squared wavefunction method, we obtain the Jacobi’s elliptic solution of the defocusing Hirota equation, and solve the related linear matrix eigenvalue problem on elliptic function background. The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique. The asymptotic dynamical
behaviors for the elliptic N-dark soliton solution as t → ±∞ are studied. Through numerical plots of the elliptic one-, two- and three-dark solitons, the amplification effect on the velocity of elliptic dark solitons, and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.

Stochastic soliton and rogue wave solutions of the nonlinear Schrödinger equation with white Gaussian noise

In this letter, we propose a novel integrable nonlinear Schrödinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schrödinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles.

A (2+1) modified KdV equation with time-dependent coefficients: exploring soliton solution via Darboux transformation and artificial neural network approach

A (2+1) modified KdV equation with time-dependent coefficients: exploring soliton solution via Darboux transformation and artificial neural network approach

On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations

We study the general elliptic KdV potentials, which can be expressed (up to adding a constant) as q_{\mathbf{p}}(z)\coloneq\sum_{j=1}^{n}m_{j}(m_{j}+1)\wp(z-p_{j}),\quad m_{j}\in\mathbb{N}. We give an elementary proof of the theorem that the singularity \mathbf{p}=( \overset{m_{1}( m_{1}+1) /2}{\overbrace{p_{1},\dots ,p_{1}}},\dots,\overset{m_{n}(m_{n}+1) /2}{\overbrace{p_{n},\dots,p_{n}}% }) is contained in the closure of the elliptic Airault–Mckean–Moser locus, which was proved previously by Treibich and Verdier in the late 1980s using purely algebro-geometric methods. Our proof is based on Darboux transformations and does not use algebraic geometry. This solves an open problem posed by Gesztesy, Unterkofler, and Weikard [Trans. Amer. Math. Soc. 358 (2006), 603–656]. Some applications are also given.

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

ANALYTICAL SOLUTIONS OF THE NONLOCAL NONLINEAR SCHRÖDINGER-TYPE EQUATIONS

In physics, nonlinear equations are applіed to characterize the varied phenomena. Usually, the nonlinear equations are presented by nonlinear partial differential equations, that can be received as conditions for the compatibility of two linear differentіal equations, named the Lax pairs. The presence of the Lax pair determines integrability for the nonlinear partial differentіal equation. Linked to this development was the realization that certаіn coherent structures, known as solіtons, which play a fundamental role in nonlinear phenomena as lattice dynamics, nonlinear optіcs, and fluіd mechanics. One of the famous equations is the nonlinear Schrödinger equation which is associated with various physical phenomena in nonlinear optics and Bose-Einstein condensates. This equation allows the Lax pair thus it is integrable. This work investigates nonlocal nonlinear Schrödinger-type equations with PT symmetry. Nonlocal nonlinear equations arise in various physical contexts as fluid dynamics, condensed matter physics, optics, and so on. We introduce the Lax pair formulation for the nonlocal nonlinear Schrödinger-type equations. The method of the Darboux transformation is applied to receive analytical solutions.

Breather–breather interactions, rational and semi-rational solutions of an integrable spin system in optical fibers

This paper investigates an integrable spin system (or Heisenberg ferromagnet-type equation or Nurshuak-IIA equation) which is an integrable generalization of the Heisenberg ferromagnet equation that raises from magneto-optical devices. The N-fold Darboux transformation of the Nurshuak-IIA equation will be constructed. With the selection of several parameters, a variety of the solutions will be produced, including breather–breather interactions, higher-order rogue waves and semi-rational solutions.

Rogue waves on the periodic background for a higher-order nonlinear Schrödinger–Maxwell–Bloch system

In this paper, we construct the rogue wave solutions on the background of the Jacobian elliptic functions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system. The Jacobian elliptic function traveling wave solutions as the seed solutions are considered. Through the approach of the nonlinearization of the Lax pair and Darboux transformation method, the rogue waves and the line rogue waves on the Jacobian elliptic functions dn and cn background are obtained, respectively.

The exploding solitons of the sine–Gordon equation

In this paper, the hodograph equivalent short pulse (HESP) equations are investigated via the Darboux transformation, we derive the soliton and positon solutions from the “seed” solutions, and then, the decomposition of the lower-order positons into single-solitons is given analytically when time T is sufficiently large. As a notable new result, we obtain the exploding soliton and positon solutions of the sine–Gordon (SG) equation from the hodograph equivalent short pulse equations.

The localized excitation on the Weierstrass elliptic function periodic background for the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation

In this paper, the linear spectral problem associated with the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (gvcKP) equation with the Weierstrass function as the external potential is investigated based on the Lamé function, from which some new localized nonlinear wave solutions on the Weierstrass elliptic ℘-function periodic background are obtained by the Darboux transformation. The degenerate solutions on the ℘-function periodic background for the gvcKP equation can be derived by taking the limits of the half-periods ω 1, ω 2 of ℘(x), whose evolution and corresponding dynamics are also discussed. The findings show that nonlinear waves on the ℘-function periodic background behave as different types of nonlinear waves in different spaces, including periodic waves, vortex solitons and interaction solutions, aiding in elucidating some physical phenomena in the related fields, such as the physical ocean and nonlinear optics.

The multi-positon and breather positon solutions for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation in this paper is the higher-order nonlinear Schrödinger equation, which can imitate the ultrashort pulses propagation in optical fibers. The modulation instability is analyzed based on the plane-wave solution. With the help of the generalized Darboux transformation, the second-, third- and fourth-order positon solutions are constructed. Furthermore, the second-, third- and fourth-order breather positon solutions are obtained, and the influences of parameters for the characteristics of solutions are analyzed.

The multi-soliton solutions of another two-component Camassa–Holm equation with Darboux transformation approach

In this paper, we develop another approach to construct the multi-soliton solutions of a two-component Camassa–Holm equation in terms of Wronskians with help of a reciprocal transformation and a gauge transformation. Its kink solution, loop solution and smooth soliton solution are presented. Then with the non-trivial limiting procedure, the solution of Camassa–Holm equation is also derived from that of two-component Camassa–Holm equation.

MKdV Equation on Time Scales: Darboux Transformation and N-Soliton Solutions

In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When x∈R and t∈R, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and N-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (R, Z, q-discrete, C).

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.