Riemann–Hilbert method and multi-soliton solutions of the Kundu-nonlinear Schrödinger equation

In this work, we study the Kundu-nonlinear Schrödinger (Kundu-NLS) equation (so-called the extended NLS equation), which can describe the propagation of the waves in dispersive media. A Lax spectral problem is used to construct the Riemann–Hilbert problem, via a matrix transformation. Based on the inverse scattering transformation, the general solutions of the Kundu-NLS equation are calculated. In the reflection-less case, the special matrix Riemann–Hilbert problem is carefully proposed to derive the multi-soliton solutions. Finally, some novel dynamics behaviors of the nonlinear system are theoretically and graphically discussed.

     

Novel solitons and higher-order solitons for the nonlocal generalized Sasa–Satsuma equation of reverse-space-time type

Nonlinear Dynamics - The general soliton solutions and higher-order soliton solutions for the nonlocal generalized Sasa–Satsuma (SS) equation of reverse-space-time type are explored. Firstly,...     

Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation

Li and Qiao studied the bifurcations and exact traveling wave solutions for the generalized two-component Camassa–Holm equation $$\begin{aligned} \left\{ \begin{array}{l} m_{t}+\sigma um_{x}-Au_{x}+2m \sigma u_{x}+3(1-\sigma )uu_{x}\\ \quad +\rho \rho _{x}=0, \\ \rho _{t} +(\rho u)_{x}=0, \end{array} \right. \end{aligned}$$ \(m=u-u_{xx}, A>0\) . They showed that there exist solitary wave solutions, cusp wave solutions, and periodic wave solutions for the equation, and their analysis focused on the bifurcations when \(\sigma >0\) . In this paper, we first complement the bifurcations when \(\sigma <0\) by following the same procedure as that of Li, and then show the existence and implicit expressions of several new types of bounded wave solutions, including solitary waves, periodic waves, compacton-like waves, and kink-like waves. In addition, the numerical simulations of the bounded wave solutions are given to show the correctness of our results.     

Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

Nonlinear Dynamics - We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrödinger equation on the line with both zero boundary condition and nonzero...     

Riemann–Hilbert approach of the Newell-type long-wave–short-wave equation via the temporal-part spectral analysis

Nonlinear Dynamics - In this paper, the Riemann–Hilbert approach is systematically established for the Newell-type long-wave–short-wave equation. Firstly, we start spectral analysis...     

New interaction of high-order breather solutions,lump solutions and mixed solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Nonlinear Dynamics - Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of...     

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

Nonlinear Dynamics - The Boussinesq equation has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave...     

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, a non-isospectral and variable-coefficient Kadomtsev–Petviashvili equation is considered using Hirota’s bilinear form and a direct assumption with...     

Weierstrass elliptic function solutions and their degenerate solutions of (2+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

Nonlinear Dynamics - Weierstrass elliptic function solutions are investigated by applying the traveling wave transformation and auxiliary equations to a (2+1)-dimensional potential...     

Rogue wave and multi-pole solutions for the focusing Kundu–Eckhaus Equation with nonzero background via Riemann–Hilbert problem method

Nonlinear Dynamics - In this work, we use the inverse scattering transform method to consider the focusing Kundu–Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the...     

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

Under investigation is a completely generalized Hirota–Satsuma–Ito equation in (2 + 1)-dimensional. Multiple lump solutions are obtained based on three test functions, including 1-, 2- and 3-order lump solutions. Subsequently, the interaction between lump wave and solitary waves, and the interaction solution between lump wave and periodic wave are studied by using the bilinear form. Final, the stability and phase velocity are investigated. In order to analyze the dynamic behavior of these solutions, some 3D plots and contour plots are given by Mathematica.

     

Bell-polynomial approach and soliton solutions for the Zhiber–Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation

Under investigation in this paper are the Zhiber?CShabat and (2+1)-dimensional Gardner equations in quantum fields, fluids and plasmas. Via the Hirota method and symbolic computation, the Bell-polynomial approach is performed to directly bilinearize those equations. For the Zhiber?CShabat equation, based on the bilinear form with an auxiliary variable, the bell-shaped soliton, upside-down bell-shaped soliton and breather-like solutions are obtained. Figures are plotted to illustrate the elastic interactions between two upside-down bell-shaped solitons and the interaction between the breather-like. As to the (2+1)-dimensional Gardner equation, bilinear form, B?cklund transformation, one- and two-shock wave solutions are derived. Amplitude-compression and amplification interactions are investigated analytically and graphically.     

Riemann–Hilbert approach and nonlinear dynamics of the coupled higher-order nonlinear Schrödinger equation in the birefringent or two-mode fiber

Nonlinear Dynamics - The multi-soliton solutions and breathers to the coupled higher-order nonlinear Schrödinger (CH-NLS) equation are derived in this work via the Riemann–Hilbert...     

Wronskian solutions and integrability for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids

Under investigation in this paper is a generalized variable-coefficient forced Korteweg–de Vries equation, which can describe the shallow-water waves, internal gravity waves, and so on. With symbolic computation, the soliton solutions in the Wronskian form are derived based on the given bilinear form. Bäcklund transformation and Lax pair for such equation are also constructed. Variable coefficients and parameters of three solitons are managed to observe the features of the solitonic propagation and interaction, e.g., the solitonic velocity, amplitude and background. Our results could be expected to benefit the relevant problems in fluids.     

Group classification and exact solutions of a generalized Emden–Fowler equation

The aim of this work is to perform a complete symmetry classification of a generalized Emden-Fowler equation. The various forms of this equation are extensively studied in the literature and they have applications in astrophysical and physiological phenomena. The classical approach of group classification and the procedure based upon the Lie algebras of low dimension are employed for classification. Exact solutions of the invariant equations are derived.     

Abundant lump and lump–kink solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Nonlinear Dynamics - By utilizing the Hirota’s bilinear form and symbolic computation, abundant lump solutions and lump–kink solutions of the new (3&nbsp;+&nbsp;1)-dimensional...     

Painlevé integrability and N-soliton solution for the variable-coefficient Zakharov–Kuznetsov equation from plasmas

With the help of symbolic computation, this paper investigates the variable-coefficient Zakharov–Kuznetsov equation which governs the two-dimensional ion-acoustic waves obliquely propagating in an inhomogeneous magnetized two-ion-temperature dusty plasma. The integrability of this model is examined through the Painlevé analysis. Via the Hirota method, the bilinear form of such model is derived. Based on the obtained bilinear form, the N-soliton solution is constructed. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis.     

Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation

Nonlinear Dynamics - In this paper, we use the bifurcation method of dynamical systems to investigate the nonlinear wave solutions of the modified Benjamin–Bona–Mahony equation. These...