Narrow Results

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.

In this study, based on the Hirota bilinear method, mixed lump-solitons, periodic lump and breather soliton solutions are derived for (2 + 1)-dimensional extended KP equation with the aid of symbolic computation. Furthermore, dynamics of these solutions are explained with 3d plots and 2d contour plots by taking special choices of the involved parameters. Through the mixed lump-soliton solutions, we observe two fusion phenomena, first from interaction of lump and single soliton and other from interaction of lump with two solitons. In both cases, lump moves gradually towards soliton and transfers energy until it completely merges with the solitons. We also observe new characteristics of periodic lump solutions and kinky breather solitons.

Self-phase modulation (SPM) induces a varying refractive index of the medium due to the optical Kerr effect. The optical waves propagation (OWP) in a medium with SPM occupied a remarkable area of research in the literature. A model equation to describe OWP in the absence of SPM was proposed very recently by Biswas–Arshed equation (BAE). This work is based on constructing the solutions that describe the waves which arise from soliton-periodic wave collisions. A variety of geometric optical wave structures are observed. Here, a transformation that allows to investigate the multi-geometric structures of OW’s result from soliton-periodic wave collisions is introduced. Chirped, conoidal, breathers, diamond and W-shaped optical waves are shown to propagate in the medium in the absence of SPM. The exact solutions of BAE are obtained by using the unified method, which was presented recently. We mention that the results found here, are completely new.

Various forms of rational solitons, breathers, rogue wave, periodic wave and multiwave for Davydov solitons in $\alpha$-helix proteins are described in this paper. We will figure out periodic wave, rogue wave, multiwave, $M$-shaped solutions, homoclinic breathers, kink cross rational solutions, periodic cross kink, periodic cross rational solitons, generalized breather, Akhmediev breather and Kuznetsov–Ma breather. We will also discuss some interactions between them. At the end, we will provide the graphical representation of our newly discovered solutions.

Different analytical solutions for Sasa–Satsuma model equation (SSM) in birefringent fibers are investigated in this research like rogue wave, lump, multi-wave, multiple breather solutions, periodic wave, breather lump, periodic cross kink, periodic cross lump, and lump with one and two kinks. By examining the SSM, we can compute the important relationship between the interaction between kink, lump with periodic wave and a variety of exact solutions. To see the behavior of effective waves, we illustrate the pictorial depiction for our retrieve analytical solutions.

This paper studies the Lax pair (LP) of the $(1+1)$-dimensional Benjamin–Bona–Mahony equation (BBBE). Based on the LP, initial solution and Darboux transformation (DT), the analytic one-soliton solution will also be obtained for BBBE. This paper contains a bunch of soliton solutions together with bright, dark, periodic, kink, rational, Weierstrass elliptic and Jacobi elliptic solutions for governing model through the newly developed sub-ODE method. The BBBE has a wide range of applications in modeling long surface gravity waves of small amplitude. In addition, we will evaluate generalized breathers, Akhmediev breathers and standard rogue wave solutions for stated model via appropriate ansatz schemes.

The generalized coupled nonlinear Schrödinger–Maxwell–Bloch system can be used to describe the propagation of optical solitons in a nonlinear light guide doped with the two-level resonant atoms. In this paper, by virtue of the Darboux transformation, bound solitons and two types of breathers for the generalized coupled nonlinear Schrödinger–Maxwell–Bloch system are generated. Furthermore, the interaction scenario between bound solitons and propagation characteristics of the breathers are discussed.

Under investigation is the complex modified Korteweg–de Vries (KdV) equation, which has many physical significance in fluid mechanics, plasma physics and so on. Via the Darboux transformation (DT) method, some breather and localized solutions are presented on two backgrounds: the continuous wave background $u_1=kexp[i(Ax+Bt)]$ and the constant background $u_2=a+ib$. Some figures are plotted to illustrate the dynamical features of those solutions.

In this paper, an eighth-order nonlinear Schrödinger equation is investigated in an optical fiber, which can be used to describe the propagation of ultrashort nonlinear pulses. Lax pair and infinitely-many conservation laws are derived to verify the integrability of this equation. Via the Darboux transformation and generalized Darboux transformation, the analytic breather and rogue wave solutions are obtained. Influence of the coefficients of operators in this equation, which represent different order nonlinearity, and the spectral parameter on the propagation and interaction of the breathers and rogue waves is also discussed. We find that (i) the periodic of the breathers decreases as the augment of the spectral parameter; (ii) the coefficients of operators change the compressibility and periodic of the breathers, and can affect the interaction range and temporal–spatial distribution of the rogue waves.

Rational solutions and hybrid solutions from N-solitons are obtained by using the bilinear method and a long wave limit method. Line rogue waves and lumps in the (2+1)-dimensional nonlinear Schrödinger (NLS) equation are derived from two-solitons. Then from three-solitons, hybrid solutions between kink soliton with breathers, periodic line waves and lumps are derived. Interestingly, after the collision, the breathers are kept invariant, but the amplitudes of the periodic line waves and lumps change greatly. For the four-solitons, the solutions describe as breathers with breathers, line rogue waves or lumps. After the collision, breathers and lumps are kept invariant, but the line rogue wave has a great change.

In this paper, a new semi-discrete coupled system which was firstly proposed by Bronsard and Pelinovsky is under investigation. Based on its known Lax pair, the infinitely-many conservation laws and discrete N-fold DT for this system are constructed. As applications, bell-shaped multi-soliton and breather solutions in terms of determinants for this system are firstly derived by means of the discrete N-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically: (1) Propagation characteristics of one-, two-, three- and four-soliton solutions are discussed from vanishing background. (2) Propagation characteristics of one- and two-breather solutions are analyzed from the plane wave background. The details of the dynamical evolutions for such soliton and breather solutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of these solitons with or without a noise in a relatively short period of time, while the evolutions exhibit obviously larger oscillations and strong instability with the increase in time. These results may be useful for understanding the propagation of orthogonally polarized optical waves in an isotropic medium and circularly polarized few-cycle pulses in Kerr media described by the coupled NLS and coupled complex mKdV equations, respectively.

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3+1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

In this paper, we study the twist of the nematic liquid crystal molecules under the applied electric field. The dynamic equation of the twisted molecules is derived. It is proved to be a kind of sine-Gordon (SG) equation. We obtain the breather solution of the equation and confirm that the deflection angles of the twisted molecules can distribute in the form of breathers. We give the relationship between the molecular deflection angle and the breather frequency, and discuss the effect of electric field on breather shape and breather frequency.

In this paper, we use the Hirota bilinear method to find the $N$-soliton solution of a $(3+1)$-dimensional generalized Kadovtsev–Petviashvili (KP) equation. Then, we obtain the $T$-order breathers of the equation, and combine the long-wave limit method to give the $M$-order lumps. Resorting to the extended homoclinic test technique, we obtain the breather-kink solutions for the equation. Last, the interaction solution composed of the $K$-soliton solution, $T$-breathers, and $M$-lumps for the $(3+1)$-dimensional generalized KP equation is constructed.

In this paper, we construct the breathers of the (3+1)-dimensional Jimbo–Miwa (JM) equation by means of the Hirota bilinear method, then based on the Hirota bilinear method with a new ansatz form, the multiple rogue wave solutions are constructed. Here, we discuss the general breathers, first-order rogue waves, the second-order rogue waves and the third-order rogue waves. Then we draw the 3- and 2-dimensional plots to illustrate the dynamic characteristics of breathers and multiple rogue waves. These interesting results will help us better reveal (3+1)-dimensional JM equation evolution mechanism.

In this paper, the coupled mixed derivative nonlinear Schrödinger equations are investigated, which govern the propagation of the femtosecond optical pulse in optical fibers. First of all, based on the soliton solutions in bilinear form, the breathers are constructed by choosing a pair of complex conjugate wave numbers. Then, the interactions between a breather and either an anti-dark soliton or a dark soliton are studied according to the existence conditions of dark and anti-dark solitons. The double-pole solution can also be obtained by a coalescence of two wave numbers. In addition, the influence of physical parameters on the phases and propagation direction of the breathers and double-pole solitons is studied by the qualitative analysis and graphical illustration.

This paper studies the Hirota–Maxwell–Bloch (H–MB) system and its nonlocal form. Based on the Darboux Transformations (DTs), for H–MB system, we present general double breathers, what is more, we take appropriate modulation frequency and position parameters to investigate the generative mechanism of rogue wave sequences and different periodic breather sequences. For nonlocal Hirota–Maxwell–Bloch (NH–MB) system, we discuss symmetry preserving and broken soliton solutions under zero background. Besides, we present nine combinations of dark and antidark soliton solutions under continuous waves background when PT-symmetry is broken.

In this paper, a ($2+1$)-dimensional generalized breaking soliton system, for the interactions of the Riemann wave with a long wave, is investigated. Via the Hirota method, bilinear forms different from those in the existing literatures are derived. $N$-soliton solutions are constructed via the Wronskian technique. Solitons with the crest curves being curvilineal are constructed, whose shape changes with the propagation. Parallel solitons have been obtained. Directions of the soliton propagation change, and speeds of the solitons are different: The higher the amplitude of the soliton is, the faster the soliton propagates. Breathers are constructed. Solutions consisting of a lump and two solitons are derived: Two solitons propagate in the same direction and the lump occurs in the region of the interaction between the two solitons.

The integrable Lakshmanan–Porsezian–Daniel (LPD) equation originating in nonlinear fiber is studied in this work via the Riemann–Hilbert (RH) approach. First, we give the spectral analysis of the Lax pair, from which an RH problem is formulated. Afterwards, by solving the special RH problem with reflectionless under the conditions of irregularity, the formula of general $N$-soliton solutions can be obtained. In addition, the localized structures and dynamic behaviors of the breathers and solitons corresponding to the real part, imaginary part and modulus of the resulting solution $r(x,t)$ are shown graphically and discussed in detail. Unlike 1- or 2-order breathers and solitons, 3-order breathers and soliton solutions rapidly collapse when they interact with each other. This phenomenon results in unbounded amplitudes which imply that higher-order solitons are not a simple nonlinear superposition of basic soliton solutions.

The coupled nonlinear Schrödinger (CNLS) system is often used to describe physical problems such as wave propagation in a birefringent optical fiber. Under investigation in this paper is the nonlocal ${𝒫}{𝒯}$-symmetric reverse-space type of CNLS system, through constructing two types of Darboux transformation (DT), we will derive a new series of nonlocal analytic solutions: (1) Single-periodic and double-periodic solutions; (2) Four different combinations about dark and anti-dark solitons, as well as the bright and dark breathers on a periodic background. Moreover, the fundamental properties and dynamical behaviors of those solutions will be discussed.