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.     

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

In this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg‐de Vries hierarchy, and nonlocal versions of the sine‐Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two‐soliton solutions of the reverse‐t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.     

Soliton Solutions for a Generalized Inhomogeneous Variable-Coefficient Hirota Equation with Symbolic Computation

Under investigation in this paper is a generalized inhomogeneous variable- coefficient Hirota equation. Through the Hirota bilinear method and symbolic computation, the bilinear form and analytic one-, two- and N-soliton solutions for such an equation are obtained, respectively. Properties of those solitons in the inhomogeneous media are discussed analytically. We get the soliton with the property that the larger the amplitude is, the narrower and slower the pulse is. Dynamics of that soliton can be regarded as a repulsion of the soliton by the external potential barrier. During the interaction of two solitons, we observe that the larger the value of the coefficient β in the equation is, the larger the distance of the two solitons is.     

Dark and antidark solitons for the defocusing coupled Sasa–Satsuma system by the Darboux transformation

In this article, we construct the $N$-fold Darboux transformation for the defocusing coupled Sasa–Satsuma system which describes the simultaneous propagation of two nonlinear waves in optical fibers with higher order effects. With the non-zero constant background as a seed, we derive the dark and antidark soliton solutions from the once-iterated formula. We find that this coupled system can exhibit the dark–dark, dark–antidark and antidark–dark vector solitons.     

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.     

Dynamics of the generalized (3 + 1)-dimensional nonlinear Schröbinger equation in cosmic plasmas

Under investigation in this paper is a generalized (3 + 1)-dimensional nonlinear Schröbinger equation with the variable coefficients, which governs the nonlinear dynamics of the ion-acoustic envelope solitons in the magnetized electron-positron-ion plasma with two-electron temperatures in space or astrophysics. Bilinear forms and Bäcklund transformations are derived through the Bell polynomials. N-soliton solutions are constructed in the form of the double Wronskian determinant and the N-th order polynomials in N exponentials. Shape and motion of one soliton have been graphically analyzed, as well as the interactions of two and three solitons. When β(t) and γ(t) are both the periodic functions of the reduced time t, where γ(t) is the loss (gain) coefficient, and β(t) means the combined effects of the transverse perturbation and magnetic field, the shape and motion of one soliton as well as the interactions of two or three solitons will occur periodically. All the interactions can be elastic with certain coefficients.     

N-soliton solutions,Bäcklund transformation and conservation laws for the integro-differential nonlinear Schröbinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain

Under investigation in this paper is an integro-differential nonlinear Schröbinger (IDNLS) equation, which is equivalent to the spin evolution equation of a classical in-homogeneous Heisenberg magnetic chain in the continuum limit. Based on the Hirota method, the bilinear form and N-soliton solution for the IDNLS equation are derived with the help of symbolic computation. Moreover, N-soliton solution for the IDNLS equation is expressed in terms of the double Wronskian and testified through the direct substitution into the bilinear form. Besides, the bilinear Bäcklund transformation and infinitely many conservation laws are also obtained for the IDNLS equation. Propagation characteristics and interaction behaviors of the solitons are discussed by analysis of such physical quantities as the soliton amplitude, width, velocity and initial phase. Interactions of the solitons are proved to be elastic through the asymptotic analysis. Effect of inhomogeneity on the interaction of the solitons is studied graphically.     

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.     

N-lump and interaction solutions of localized waves to the (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid

The present article deals with M-soliton solution and N-soliton solution of the (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump, and their interactions, which have been investigated by the approach of the long-wave limit. Mainly, by choosing the specific parameter constraints in the M-soliton and N-soliton solutions, all cases of the one breather or one lump can be captured from the two, three, four, and five solitons. In addition, the performances of the mentioned technique, namely, the Hirota bilinear technique, are substantially powerful and absolutely reliable to search for new explicit solutions of nonlinear models. Meanwhile, the obtained solutions are extended with numerical simulation to analyze graphically, which results in localized waves and their interaction from the two-, three-, four-, and five-soliton solutions profiles. They will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.     

Dark solitonic excitations and collisions from a fourth-order dispersive nonlinear Schrödinger model for the alpha helical protein

Recent protein observations motivate the dark-soliton study to explain the energy transfer in the proteins. In this paper we will investigate a fourth-order dispersive nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. Painlevé analysis is performed to prove the equation is integrable. Through the introduction of an auxiliary function, bilinear forms and dark N-soliton solutions are constructed with the Hirota method and symbolic computation. Asymptotic analysis on the two-soliton solutions indicates that the soliton collisions are elastic. Decrease of the coefficient of higher-order effects can increase the soliton velocities. Graphical analysis on the two-soliton solutions indicates that the head-on collision between the two solitons, overtaking collision between the two solitons and collision between a moving soliton and a stationary one are all elastic. Collisions among the three solitons are all pairwise elastic.     

Some properties of the spinor soliton

In this paper, the solitons of nonlinear Dirac equation are discussed in detail, and several functions which reflect their characteristics are computed. The numerical results show that, the nonlinear Dirac equation has only finite meaningful solitons, and these solitons have 1/2-spin and positive mass; the spinor soliton has two kinds of parity states, and each parity state has two kinds of energy states; the larger the self-coupling coefficientw, the more the excitation states, and ifw is less than a critical value, then the meaningful soliton does not exist. These properties may have relations with some fundamental particles.     

Soliton interactions and Yang–Baxter maps for the complex coupled short-pulse equation

The complex coupled short-pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so-called self-symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long-time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well-known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.     

Exact soliton solutions of a D-dimensional nonlinear Schr?dinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schr?dinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

APPLICATIONS OF THE P-R SCHEME FOR GENERALIZED NONLINEAR SCHRODINGER EQUATIONS IN SOLVING SOLITON SOLUTIONS

Spatial soliton solutions of a class of generalized nonlinear Schrodinger equations in N-space are discussed analytically and numerically. This achieved using a traveling wavemethod to formulate one-soliton solution and the P-R method is employed to the numerlcal solutions and the interactions between the solirons for the generalized nonlinear systems in Z-pace.The results presented show that the soliton phenomena are characteristics associated with the nonlinearhies of the dynamical systems.     

Dark soliton control in inhomogeneous optical fibers

Analytic two-dark soliton solutions for a variable–coefficient nonlinear Schrödinger equation are obtained via modified Hirota method. Parallel solitons are observed and soliton control such as the soliton compression is realized with different group velocity dispersion profiles. Besides, soliton interactions are investigated with the interaction distance being adjusted. In addition, soliton repulsive structures as well as attractive ones are obtained with exponential dispersion profile. Results in our research may be useful for the soliton control in inhomogeneous optical fibers, which will be a benefit to the realistic optical communication systems.     

Numerical simulations of the improved Boussinesq equation

In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second‐and third‐order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L_∞ error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Solutions of Kadomtsev–Petviashvili equation with power law nonlinearity in 1+3 dimensions

This paper studies the solution of the Kadomtsev–Petviasvili equation with power law nonlinearity in 1+3 dimensions. The Lie symmetry approach as well as the extended tanh‐function and G′/G methods are used to carry out the analysis. Subsequently, the soliton solution is obtained for this equation with power law nonlinearity. Both topological as well as non‐topological solitons are obtained for this equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Dark–dark and dark–bright soliton interactions in the two-component defocusing nonlinear Schrödinger equation

We use the Inverse Scattering Transform machinery to construct multisoliton solutions to the 2-component defocusing nonlinear Schrödinger equation. Such solutions include dark–dark solitons, which have dark solitonic behaviour in both components, as well as dark–bright soliton solutions, with one dark and one bright component. We then derive the explicit expressions of two soliton solutions for all possible cases: two dark–dark solitons, two dark–bright solitons, and one dark–dark and one dark–bright soliton. Finally, we determine the long-time asymptotic behaviours of these solutions, which allows us to obtain explicit expressions for the shifts in the phases and in the soliton centers due to the interactions.