N-fold Darboux Transformation Research Articles

Determinant representation of Darboux transformation for the (2+1)-dimensional nonlocal nonlinear Schrödinger-Maxwell-Bloch equations

In this paper, we mainly investigate two-fold Darboux transformation and N-fold Darboux transformation in terms of ratio of ordinary determinant for the (2+1)-dimensional nonlocal nonlinear Schrödinger-Maxwell-Bloch equations with reverse-space. It should be mentioned that the questions of one-fold Darboux transformation addressed, results of one-fold Darboux transformation proved, as well as style of analysis borrow heavily, but the determinant representation involved to cope with higher-order Darboux transformation is nontrivial. By utilizing symbolic computation, some explicit solutions including periodic waves solutions, soliton solutions and complexion solutions are obtained for two-fold Darboux transformation. Choosing particular parameters, figures are plotted to show the propagations and interactions of various types of solitons, such as periodic waves, bright soliton, dark soliton, antidark soliton, two-peak bright soliton, two-peak dark soliton and two-peak antidark soliton.

DARBOUX TRANSFORMATIONS, MULTISOLITONS, BREATHER AND ROGUE WAVE SOLUTIONS FOR A HIGHER-ORDER DISPERSIVE NONLINEAR SCHRÖDINGER EQUATION

<abstract> In this paper, dynamic of a higher-order dispersive nonlinear Schrö-dinger equation is investigated. Firstly, we obtain the determinant representation of the N-fold Darboux transformations of the Schrödinger equation. Then based on the above analysis, we get the one-soliton, two-soliton and the breather wave solution. Furthermore, the first-order rogue wave is derived by means of a Taylor expansion of the breather wave. Finally, by selecting some special parameters and drawing the 3-D and 2-D graphs to better describe the dynamic traits of those solutions. </abstract>

Multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation

In this letter, we investigate multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation over a nonzero background. First, we obtain 2n-soliton solutions with a nonzero background via n-fold Darboux transformation, and find that these soliton solutions will appear in pairs. Particularly, 2n-soliton solutions consist of n ‘bright’ solitons and n ‘dark’ solitons. This phenomenon implies a new form of integrability: even integrability. Then interactions between solitons with even numbers and breathers are studied in detail. To our best knowledge, a novel nonlinear superposition between a kink and 2n-soliton is also generated for the first time. Finally, interactions between some different smooth positons with a nonzero background are derived.

Matrix Spectral Problems and Integrability Aspects of the Błaszak-Marciniak Lattice Equations

A method to derive the matrix spectral problems of the Blaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Blaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Blaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the N-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviours of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard way

Darboux transformation of the generalized mixed nonlinear Schrödinger equation revisited.

In this paper, we present an explicit Darboux transformation of the generalized mixed nonlinear Schrödinger (GMNLS) equation. The compact determinant representation of the n-fold Darboux transformation of the GMNLS equation is constructed and the nth-order solution is built. We further prove that only the even-fold Darboux transformation and the even-order solution of the GMNLS equation can, respectively, be reduced to the Darboux transformation and solution of the Kundu-Eckhaus equation. Furthermore, two different kinds of explicit one-soliton solutions of the GMNLS equation are constructed and discussed.

Higher-order rogue wave solutions for a generalized Hirota-Maxwell-Bloch system in the erbium-doped fibers

Under investigation in this paper is a generalized Hirota-Maxwell-Bloch System, which describes the propagation of optical solitons in the erbium-doped medium. With symbolic computation, the generalized N-fold Darboux transformation (GDT) will be constructed. As the application of the GDT, some types of higher-order rogue waves will be presented accurately by choosing different parameters and dynamic behaviors of those rogue waves will be analyzed.

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12071042 and 61471406), the Beijing Natural Science Foundation, China (Grant No. 1202006), and Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP-B201704).

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

Degenerate solutions for the spatial discrete Hirota equation

In this paper, some degenerate solutions of the spatial discrete Hirota equation are constructed via the degenerate idea of positon solution. Under the zero seed solution, the n-positon is obtained by N-fold degenerate Darboux transformation (DT). The degenerate DT is taking the degenerate limit $$\lambda _{j}\rightarrow \lambda _{1}$$ for the eigenvalues $$\lambda _{j}(j=1,2, 3, \ldots , N)$$ of N-fold DT and then performing the high-order Taylor expansion near $$\lambda _{1}$$ . Considering the universal Darboux transformation, breather is obtained from the nonzero seed. Then, a new type of breather solution can be produced by using the same degenerated method and higher-order Taylor expansion for eigenvalues in determinant expression of breather solution. The explicit determinants of breather-positon solution and positon solution are constructed, respectively, and the complicated and significant dynamics of low-order solution are also revealed.

Lax pairs, infinite conservation laws, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation

Calogero-Bogoyavlenskii-Schiff-type (CBS-type) equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Under investigation in this paper is a combined CBS-type equation. Lax pairs in the differential and matrix forms are derived respectively. Infinite conservation laws, which are different from those in the existing literatures, and n-fold Darboux transformation are constructed through the matrix-form Lax pair, where n is a positive integer. Bilinear forms are constructed. Parallel solitons can be derived from the bilinear forms, which are caused by a constraint in the linearization process. Waveforms for the parallel solitons have the no superposition, nonlinear superposition and linear superposition forms. Bell-to-anti-bell-shaped solitons and oblique solitonic interactions are discovered via the n-fold Darboux transformation. Each bell-shaped asymptotic soliton of the bell-to-anti-bell-shaped soliton evolves into the anti-bell-shaped one.

A generalized complex mKdV equation: Darboux transformations and explicit solutions

A new generalized complex modified Korteweg–de Vries equation associated with a 3 × 3 matrix spectral problem is proposed by resorting to the zero-curvature equation. Based on the gauge transformations between the Lax pairs, a Darboux transformation for the generalized complex modified Korteweg–de Vries equation is constructed, from which the corresponding N-fold Darboux transformations are derived in terms of both iterative technique and determinants. As an application of the resulting Darboux transformations, explicit solutions, like one-soliton, two-soliton, and three-soliton solutions, first-order breather and second-order breather solutions, for the generalized complex modified Korteweg–de Vries equation are obtained.

N-fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation

In this paper, the N-fold Darboux transformation (DT) of the combined Toda lattice and relativistic Toda lattice equation is constructed in terms of determinants. Comparing with the usual 1-fold DT of equations, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications of the N-fold DT, we derive two kinds of N-fold explicit exact solutions from two different seed solutions and plot the figures with properly parameters to illustrate the propagation of solitary waves. What’s more, we present the relationships between the structures of exact solutions parameters with $$N=1$$ , from which we find the 1-fold solutions may be one soliton solutions or periodic solutions and the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The results in this paper might be helpful for interpreting certain physical phenomena.

A pair of modified short pulse equations and its two-component system in nonlinear media

A pair of modified short pulse equations is introduced and its integrable two-component system is discussed. The relations between the short pulse equations and the modified short pulse equations are established. The parametric representations of various solutions are derived by constructing their N-fold Darboux transformations. In particular, multiple smooth solitons, cuspon solitons, loop solitons and breather solutions of the two-component modified short pulse system are discussed in detail and interesting dynamical interactions of its solutions are illustrated with figures.

Soliton elastic interactions and dynamical analysis of a reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon

Under investigation in this paper is a discrete reduced integrable nonlinear Schrodinger system on a triangular-lattice ribbon, which may have some prospective applications in modern nanoribbon. First, we construct the infinitely many conservation laws and discrete N-fold Darboux transformation for this system based on its known Lax pair. Then bright–bright multi-soliton and breather solutions in terms of determinants are obtained by means of the resulting Darboux transformation. Moreover, we investigate soliton interactions through asymptotic analysis and analyze some important physical quantities such as amplitudes, wave numbers, wave widths, velocities, energies and initial phases. Finally, the dynamical evolution behaviors are discussed via numerical simulations. It is found that soliton interactions in this system are elastic, and their evolutions are stable against a small noise in a short period of time. Results obtained in this paper may have some prospective applications for understanding some physical phenomena.

On two new types of modified short pulse equation

A complex modified short pulse (mSP) equation and a nonlocal mSP equation are presented. The integrability and the covariant property of both new equations are shown by constructing their corresponding Lax pair and Darboux transformation, respectively. For the complex mSP equation, some multiple smooth soliton, cuspon soliton, loop soliton, breather and rogue wave solutions are constructed by N-fold Darboux transformation. For the nonlocal mSP equation, some unstable soliton and stable soliton solutions are obtained. The dynamical characteristics of the obtained solutions are shown through some figures.

Breather solutions and breather-to-soliton conversions for the (2+1)-dimensional complex modified Korteweg-de Vries and Maxwell-Bloch equations

In this paper, we will focus on the (2+1)-dimensional complex modified Korteweg-de Vries and Maxwell-Bloch (cmKdVMB) equations. According to the correlative Lax pair, the N-fold Darboux Transformations(DT) will be constructed. By virtue of the DT obtained, two different kinds of breather solutions and breather-to-soliton conversions will be derived from the nonvanishing continuous wave (cw) solutions. Meanwhile, the dynamic features of those solutions will be graphically analyzed.

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev–Petviashvili equation

For a variable coefficient Kadomtsev–Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painlevé analysis. The N-fold binary Darboux transformation is presented in a compact form. As an application, the multi-lump, higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained. Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.

Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity

In this work, we consider a nonlinear Schrodinger equation with stimulated Raman scattering and quintic nonlinearity, which works as a model for the propagation of ultrashort pulse in optical fiber and also corresponds to one-dimensional anisotropic Heisenberg ferromagnetic spin chain. Introducing a Galilean transformation, we transform the model into a fifth-order complex modified KdV equation, the integrability of which has been considered in the AKNS technique framework. The N-fold Darboux transformation for the model is derived in terms of the gauge transformation of the associated $$3 \times 3$$ matrix spectral problem. As applications, abundant intriguing types of nonlinear waves are obtained in zero and nonzero boundary conditions. The dynamical evolution of these nonlinear waves can be well controlled under stimulated Raman scattering and quintic nonlinearity management. Beside that, we also find some novel and interesting conversion phenomena, where a high-order term plays a pivotal role.

Soliton interactions and their dynamics in a higher-order nonlinear self-dual network equation

Under investigation in this paper is a higher-order nonlinear self-dual network equation, which may simulate the wave propagation in a ladder type electric circuit. By means of the N-fold Darboux transformation and symbolic computation, the N-soliton solutions in determinant form are obtained. Based on the asymptotic and graphic analysis, the elastic interaction phenomena between/among two-, three- and four-soliton solutions are discussed, and some important physical quantities are accurately analyzed. Numerical simulations are used to explore the dynamical stability of one- and two-soliton solutions. Results might be helpful for understanding the propagation and interaction properties of electrical signals in a ladder type nonlinear self-dual network.

N-fold Darboux transformation of the two-component Kundu–Eckhaus equations and non-symmetric doubly localized rogue waves

The two-component Kundu–Eckhaus (KE) equations were introduced for the first time in 1999. Very recently, the two-component KE equations considered as a model of describing the effect of quintic nonlinearity on the ultra-short optical pulse propagation in non-Kerr media have been intensively studied. In this paper, we construct an analytical and explicit representation of the Darboux transformation (DT) for the two-component KE equations. Compared with the DT constructed by researchers before, the DT here is expressed by the initial eigenfunctions, spectral parameters, and ‘seed’ solution. As applications of DT, the explicit expressions of non-symmetric rogue wave of two-component KE equations and KE equation are displayed, and the differences between the non-symmetric and symmetric rogue wave for the KE equation are discussed in detail.

Dynamics of localized wave solutions for a higher-order Ablowitz-Ladik equation

It is an important research topic to study diverse local wave interaction phenomena in nonlinear evolution equations, especially for the semi-discrete nonlinear lattice equations, there is little work on their diverse local wave interaction solutions due to the complexity and difficulty of research. In this paper, a semi-discrete higher-order Ablowitz-Ladik equation is investigated via the generalized &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$(M, N-M)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-fold Darboux transformation. With the aid of symbolic computation, diverse types of localized wave solutions are obtained starting from constant and plane wave seed background. Particularly, for the case &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$M=N$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the generalized &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$(M, N-M)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-fold Darboux transformation may reduce to the &lt;i&gt;N&lt;/i&gt;-fold Darboux transformation which can be used to derive multi-soliton solutions from constant seed background and breather solutions from plane wave seed background, respectively. For the case &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$M=1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M5.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M5.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the generalized &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$(M, N-M)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M6.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M6.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-fold Darboux transformation reduce to the generalized &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$(1, N-1)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M7.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M7.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-fold one which can be used to obtain rogue wave solutions from plane wave seed background. For the case &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$M=2$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the generalized &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$(M, N-M)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-fold Darboux transformation reduce to the generalized &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$(2, N-2)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M10.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191235_M10.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-fold one which can be used to give mixed interaction solutions of one-breather and first-order rogue wave from plane wave seed background. To study the propagation characteristics of such localized waves, the numerical simulations are used to explore the dynamical stability of such obtained solutions. Results obtained in the present work may be used to explain related physical phenomena in nonlinear optics and relevant fields.