Positon Solutions Research Articles

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.

Nonlocal M-component nonlinear Schrödinger equations: Bright solitons, energy-sharing collisions, and positons.

The general set of nonlocal M-component nonlinear Schrödinger (nonlocal M-NLS) equations obeying the PT-symmetry and featuring focusing, defocusing, and mixed (focusing-defocusing) nonlinearities that has applications in nonlinear optics settings, is considered. First, the multisoliton solutions of this set of nonlocal M-NLS equations in the presence and in the absence of a background, particularly a periodic line wave background, are constructed. Then, we study the intriguing soliton collision dynamics as well as the interesting positon solutions on zero background and on a periodic line wave background. In particular, we reveal the fascinating shape-changing collision behavior similar to that of in the Manakov system but with fewer soliton parameters in the present setting. The standard elastic soliton collision also occurs for particular parameter choices. More interestingly, we show the possibility of such elastic soliton collisions even for defocusing nonlinearities. Furthermore, for the nonlocal M-NLS equations, the dependence of the collision characteristics on the speed of the solitons is analyzed. In the presence of a periodic line wave background, we notice that the soliton amplitude can be enhanced significantly, even for infinitesimal amplitude of the periodic line waves. In addition to these solutions, by considering the long-wavelength limit of the obtained soliton solutions with proper parameter constraints, higher-order positon solutions of the nonlocal M-NLS equations are derived. The background of periodic line waves also influences the wave profiles and amplitudes of the positons. Specifically, the positon amplitude can not only be enhanced but also be suppressed on the periodic line wave background of infinitesimal amplitude.

Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11775121 and 11435005), and the K. C. Wong Magna Fund in Ningbo University.

Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent “phase shifts” of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.

Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schrödinger equation

Based on the degenerate Darboux transformation, the $n$-order smooth positon solutions for the derivative nonlinear Schr\"{o}dinger equation are generated by means of the general determinant expression of the $N$-soliton solution, and interesting dynamic behaviors of the smooth positons are shown by the corresponding three dimensional plots in this paper. Furthermore, the decomposition process, bent trajectory and the change of the phase shift for the positon solutions are discussed in detail. Additional, three kinds of mixed solutions, namely (1) the hybrid of one-positon and two-positon solutions, (2) the hybrid of two-positon and two-positon solutions, and (3) the hybrid of one-soliton and three-positon solutions are presented and their rather complicated dynamics are revealed.

The n-fold Darboux transformation for the Kundu-Eckhaus equation and dynamics of the smooth positon solutions

In this paper, we completely discuss the derivation of Darboux transformation (DT) of the Kundu-Eckhaus (KE) equation by a direct way and an indirect way respectively. The compact determinant representation of formula of n-fold DT for the KE equation is constructed and nth-order solution is presented by the DT. Furthermore, we obtain the formula of the n-positon solution for the KE equation by using the Taylor expansion respect to degenerate eigenvalues λ2k−1→λ1(k=1,2,…,n). Especially, the exact expressions of soliton and positon solutions are obtained by the corresponding formulas. The dynamics of the smooth positons of the KE equation are discussed by using the decomposition of the modulus square, which can be used to depict approximately the trajectories and time-dependent ‘phase shifts’ of positons after the collision.

Riemann–Hilbert method for the Wadati–Konno–Ichikawa equation: N simple poles and one higher-order pole

Abstract We present a Riemann–Hilbert (RH) method directly by using the inverse scattering method (ISM) for Wadati–Konno–Ichikawa equation (WKIE) i q t + q 1 + | q | 2 x x = 0 . The RH problem is related to two cases of scattering data: N simple poles and one N th order pole. Under the reflection-less situation, we solve the RH problem and obtain the formulae of N th order soliton and positon solutions in the form of determinants. As applications, the first-order soliton and the second-order positon solutions are displayed in analytical and graphical ways. For the first-order soliton, it displays analytic form, bursting form, and loop form. For the second-order positon solution, an interesting inelastic phenomenon is observed that a singular positon solution including an analytic soliton and a loop soliton is split into an analytic soliton and a bursting soliton after the collision.

Smooth Positons of the Second-Type Derivative Nonlinear Schrödinger Equation**Supported by the National Natural Science Foundation of China under Grant No. 11671219, the K. C. Wong Magna Fund in Ningbo University, and the Natural Science Foundation of Zhejiang Province under Grant No. LZ19A010001

We construct the soliton solution and smooth positon solution of the second-type derivative nonlinear Schrödinger (DNLSII) equation. Additionally, we present a detailed discussion about the decomposition of the positon solution, and display its approximate orbits and variable “phase shift”. The second and third order breather-positon solutions are also constructed.

Dynamics of soliton solutions of the nonlocal Kundu-nonlinear Schrödinger equation.

In this paper, we investigate the nonlocal Kundu-nonlinear Schrödinger (Kundu-NLS) equation, which can be obtained from the reduction of the coupled Kundu-NLS system. Based on the analysis of the eigenfunctions, a Riemann-Hilbert problem is constructed to derive the N-soliton solutions of the coupled Kundu-NLS system. The N-soliton solutions of the nonlocal Kundu-NLS equation are then deduced with properly chosen symmetry relations on the scattering data. The dynamics of the solitons in the nonlocal Kundu-NLS equation are explored. The impact of the gauge function on the solitons is displayed for one-solitons. Compared with the dynamics of the two-solitons in the local Kundu-NLS equation, the two-solitons in the nonlocal Kundu-NLS equation display many differences. The repeated collapsing is a common feature of the singular solitons, and it seems that some of them are not the superposition of one-solitons. The singular solitons exhibit various behaviors in different eigenvalue configurations in the spectral space. Besides that, three kinds of bounded solutions are presented according to these eigenvalue configurations. In addition, two kinds of degenerate solutions are presented, and in particular, the positon solutions are discussed in detail. The decomposition of the positon solutions is analyzed and their trajectories are given approximately.

Dynamics of the Smooth Positons of the Wadati-Konno-Ichikawa Equation* *Supported by the National Natural Science Foundation of China under Grant No. 11671219, the K. C. Wong Magna Fund in Ningbo University

We discuss a modified Wadati-Konno-Ichikawa (mWKI) equation, which is equivalent to the WKI equation by a hodograph transformation. The explicit formula of degenerated solution of mWKI equation is provided by using degenerate Darboux transformation with respect to the eigenvalues, which yields two kinds of smooth solutions possessing the vanishing and nonvanishing boundary conditions respectively. In particular, a method for the decomposition of modulus square is operated to the positon solution, and the approximate orbits before and after collision of positon solutions are displayed explicitly.

WRONSKIAN SOLUTIONS OF (2+1) DIMENSIONAL NON-LOCAL ITO EQUATION

In this work, the Wronskian determinant technique is performed to(2+1)-dimensional non-local Ito equation in the bilinear form. First, we obtainsome su¢ cient conditions in order to show Wronskian determinant solves the(2+1)-dimensional non-local Ito equation. Second, rational solutions, solitonsolutions, positon solutions, negaton solutions and their interaction solutionswere deduced by using the Wronskian formulations

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.

Smooth positon solutions of the focusing modified Korteweg–de Vries equation

The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo\-di\-fied Kor\-te\-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues $\lambda_{j}$ and the corresponding eigenfunctions of the associated Lax equation. The nonsingular $n$-positon solutions of the focusing mKdV equation are obtained in the special limit $\lambda_{j}\rightarrow\lambda_{1}$, from the corresponding $n$-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the $n$-positon solution into $n$ single-soliton solutions, the trajectories, and the corresponding "phase shifts" of the multi-positons are also investigated.

Dynamics of the smooth positons of the complex modified KdV equation

We provide the explicit formulae of the smooth positon solutions of the complex modified KdV (mKdV) equation using degenerate Darboux transformation with respect to the eigenvalues. The dynamics of the smooth positons of the complex mKdV are discussed in details using the method, i.e. decomposition of the modulus square. For this kind solution, we show explicitly the decomposition, bent trajectory and variable ‘phase shift’ after collision, which are remarkably different from the singular positons of the real-valued mKdV equation.

New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.

Darboux transformation and Rogue waves of the Kundu-nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n-fold Darboux transformation. From known solution Q, the determinant representation of n-th new solutions of Q[n] are obtained by the n-fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third-order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.

Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation

In this paper, we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch (IH-MB) equations which are governed by femtosecond pulse propagation through inhomogeneous doped fibre. The determinant representation of Darboux transformation is used to derive soliton solutions, positon solutions to the IH-MB equations.

Darboux Transformation and Solutions of the Two-Component Hirota—Maxwell—Bloch System

We derive the n-fold Darboux transformation of the two-component Hirota and Maxwell—Bloch (TH-MB) equations and its determinant representation. Using Darboux determinant representation, we provide soliton solutions, positon solutions of the TH-MB equations.

On rogue wave in the Kundu-DNLS equation

In this paper, the determinant representation of the n-fold Darboux transformation (DT) of the Kundu-DNLS equation is given. Based on our analysis, the soliton solutions, positon solutions and breather solutions of the Kundu-DNLS equation are given explicitly. Further, we also construct the rogue wave solutions which are given by using the Taylor expansion of the breather solution. Particularly, these rogue wave solutions possess several free parameters. With the help of these parameters, these rogue waves constitute several patterns, such as fundamental pattern, triangular pattern, circular pattern.

Nonlinear waves of the Hirota and the Maxwell—Bloch equations in nonlinear optics

In this paper, considering the Hirota and the Maxwell—Bloch (H-MB) equations which are governed by femtosecond pulse propagation through a two-level doped fiber system, we construct the Darboux transformation of this system through a linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.

Rogue waves of the Hirota and the Maxwell-Bloch equations

In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch (H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fiber and further generalize it to the matrix form of the n-fold Darboux transformation of this system. This n-fold Darboux transformation implies the determinant representation of nth solutions of (E([n]),p([n]),η([n])) generated from the known solution of (E,p,η). The determinant representation of (E([n]),p([n]),η([n])) provides soliton solutions, positon solutions, and breather solutions (both bright and dark breathers) of the H-MB system. From the breather solutions, we also construct a bright and dark rogue wave solution for the H-MB system, which is currently one of the hottest topics in mathematics and physics. Surprisingly, the rogue wave solution for p and η has two peaks because of the order of the numerator and denominator of them. Meanwhile, after fixing the time and spatial parameters and changing two other unknown parameters α and β, we generate a rogue wave shape.