Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Bäcklund transformation of variable-coefficient Boiti–Leon–Manna–Pempinelli equation

In this letter, we discuss a variable-coefficient Boiti–Leon–Manna–Pempinelli equation. We present its soliton solution and derive its new bilinear Bäcklund transformation through Bell polynomial technique and bilinear method. Finally, we show the variable-coefficient Boiti–Leon–Manna–Pempinelli equation is completely integrable.     

Soliton Interactions for the Three‐Coupled Discrete Nonlinear Schrödinger Equations in the Alpha Helical Proteins

Three‐coupled discrete nonlinear Schrödinger equations, which describe the dynamics of the three hydrogen bonding spines in the alpha helical proteins with the interspine coupling at the discrete level, are investigated. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation of those equations. Propagation characteristics and interactions of the bound‐state solitons are discussed. Bound states of two and three bright solitons arise when all of them propagate in parallel. Elastic interaction between the bound‐state solitons and one bright soliton is given. Increase of the dipole‐dipole interaction energy can lead to the increase of the soliton velocity, that is, the one‐interaction period becomes shorter.     

The Derivative Yajima–Oikawa System: Bright,Dark Soliton and Breather Solutions

In this paper, we study the derivative Yajima–Oikawa (YO) system which describes the interaction between long and short waves (SWs). It is shown that the derivative YO system is classified into three types which are similar to the ones of the derivative nonlinear Schrödinger equation. The general N ‐bright and N ‐dark soliton solutions in terms of Gram determinants are derived by the combination of the Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Particularly, it is found that for the dark soliton solution of the SW component, the magnitude of soliton can be larger than the nonzero background for some parameters, which is usually called anti‐dark soliton. The asymptotic analysis of two‐soliton solutions shows that for both kinds of soliton only elastic collision exists and each soliton results in phase shifts in the long and SWs. In addition, we derive two types of breather solutions from the different reduction, which contain the homoclinic orbit and Kuznetsov–Ma breather solutions as special cases. Moreover, we propose a new (2+1)‐dimensional derivative Yajima–Oikawa system and present its soliton and breather solutions.     

Multiple soliton solutions for (2 + 1)‐dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations

Multiple soliton solutions for the (2 + 1)‐dimensional Sawada–Kotera and the Caudrey–Dodd–Gibbon equations are formally derived. Moreover, multiple singular soliton solutions are obtained for each equation. The simplified form of Hirota's bilinear method is employed to conduct this analysis. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Bilinear Form and N‐Shock‐Wave Solutions for a (2+1)‐Dimensional Breaking Soliton Equation in Certain Fluids with the Bell Polynomials and Auxiliary Function

In this paper, we will investigate a (2+1)‐dimensional breaking soliton (BS) equation for the (2+1)‐dimensional collision of a Riemann wave with a long wave in certain fluids. Using the Bell polynomials and an auxiliary function, we derive a new bilinear form for the (2+1)‐dimensional BS equation, which is different from those in the previous literatures. One‐, two‐ and N‐shock‐wave solutions are obtained with the Hirota method and symbolic computation. One shock wave is found to be able to stably propagate. Two shock waves are observed to have the parallel collision, oblique collision, and stable propagation of the V‐type structure. In addition, we present the collision between one shock wave and V‐type structure, and the collision between two V‐type structures.     

Multi‐component integrable couplings for the Ablowitz–Kaup–Newell–Segur and Volterra hierarchies

A kind of N × N non‐semisimple Lie algebra consisting of triangular block matrices is used to generate multi‐component integrable couplings of soliton hierarchies from zero curvature equations. Two illustrative examples are made for the continuous Ablowitz–Kaup–Newell–Segur hierarchy and the semi‐discrete Volterra hierarchy, together with recursion operators. Copyright © 2014 John Wiley & Sons, Ltd.     

New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

In this paper, I introduce a class of super Bell polynomials, which are found to play an important role in the characterization of super supersymmetric equations. An effective approach based on the use of the super Bell polynomials is developed to systematically investigate the bilinearization, Bäcklund transformation, and Lax pair for supersymmetric equations. I take a supersymmetric two‐boson equation to illustrate this procedure. A new bilinear Bäcklund transformation and a Lax pair with both fermionic and bosonic parameters are given. In addition, a kind of exact solitons for the equation are further constructed with the help of the bilinear Bäcklund transformation.     

Interaction among a lump,periodic waves,and kink solutions to the fractional generalized CBS‐BK equation

The Hirota bilinear method is prepared for searching the diverse soliton solutions for the fractional generalized Calogero‐Bogoyavlenskii‐Schiff‐Bogoyavlensky‐Konopelchenko (CBS‐BK) equation. Also, the Hirota bilinear method is used to finding the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and multi‐kink soliton solutions will be investigated. Also, the solitary wave, periodic wave, and cross‐kink wave solutions will be examined for the fractional gCBS‐BK equation. The graphs for various fractional order α are plotted to contain 3D plot, contour plot, density plot, and 2D plot. We construct the exact lump and interaction among other types solutions, by solving the under‐determined nonlinear system of algebraic equations for the associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed. The existence conditions are employed to discuss the available got solutions.     

Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H¹‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L²‐norm in the previous literature. Furthermore, the global superconvergence results in H¹‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis.     

Darboux Transformations and Global Explicit Solutions for Nonlocal Davey–Stewartson I Equation

For the nonlocal Davey–Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like other nonlocal equations, many solutions of this equation may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of t , respectively, by a Darboux transformation of degree n are global solutions of the nonlocal Davey–Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have n peaks, and “dark cross soliton” solutions when the seed solution is an exponential function of t , in the sense that they are bounded and their norms change fast along some crossing straight lines.     

Integrability and soliton interaction of a resonant nonlinear Schrödinger equation via binary Bell polynomials

Under investigation in this paper is a resonant nonlinear Schrödinger equation for the response of a hypothetical resonance medium to an action of a quasimonochromatic wave or the propagation of one-dimensional long magnetoacoustic waves in a cold collisionless plasma subject to a transverse magnetic field. Binary Bell polynomials are employed to derive the bilinear form, Bäcklund transformation (BT) and Lax pair in the 3×3 matrix form. Two sets of the binary Bell polynomials are considered. Infinite conservation laws are also constructed from the BT in the binary-Bell-polynomial form. Moreover, two-soliton solutions are obtained through the Hirota method. Finally, the regular, intermediate-state and resonant soliton interactions are analyzed under certain conditions.     

On the integrability of the (1+1)-dimensional and (2+1)-dimensional Ito equations

Under investigation in this paper are the (1+1)-dimensional and (2+1)-dimensional Ito equations. With the help of the Bell polynomials method, Hirota bilinear method and symbolic computation, the bilinear representations, N-soliton solutions, bilinear Bäcklund transformations and Lax pairs of these two equations are obtained, respectively. In particular, we obtain a new bilinear form and N-soliton solutions of the (2+1)-dimensional Ito equation. The bilinear Bäcklund transformation and Lax pair of the (2+1)-dimensional Ito equation are also obtained for the first time. Copyright © 2014 John Wiley & Sons, Ltd.     

General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations

N‐dark–dark solitons in the integrable coupled NLS equations are derived by the KP‐hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed‐nonlinearity case, two dark–dark solitons can form a stationary bound state.     

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

We consider the fully parity‐time (PT) symmetric nonlocal (2 + 1)‐dimensional nonlinear Schrödinger (NLS) equation with respect to x and y. By using Hirota's bilinear method, we derive the N‐soliton solutions of the nonlocal NLS equation. By using the resulting N‐soliton solutions and employing long wave limit method, we derive its nonsingular rational solutions and semi‐rational solutions. The rational solutions act as the line rogue waves. The semi‐rational solutions mean different types of combinations in rogue waves, breathers, and periodic line waves. Furthermore, in order to easily understand the dynamic behaviors of the nonlocal NLS equation, we display some graphics to analyze the characteristics of these solutions.     

Dynamics of Strongly Nonlinear Kinks and Solitons in a Two‐Layer Fluid

Perturbation theory is developed for interaction of strongly nonlinear solitary waves close to the limiting, tabletop solitons (Π‐solitons). The method is based on representing each soliton as a compound of two kinks so that the interaction of N solitons is treated as the interaction of 2N kinks. As an example the Miyata–Choi–Camassa equations for a two‐layer fluid is considered. Equations for kink coordinates are obtained and analyzed. Some nontrivial features of two‐soliton interaction characteristic of the strongly nonlinear case are established.