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This study investigates the nonlinear wave equation in soliton theory, specifically focusing on the new generalized (3+1)-dimensional P -type equation. By employing the Hirota bilinear method, we successfully derive the dynamics of multiwave solutions with trigonometric and hyperbolic functions such as one-soliton, two-soliton, three-soliton, Kuznetsov–Ma breather soliton, Homoclinic breather soliton, M-shaped soliton and W-shaped soliton. To visually represent the obtained solutions, we depict them through 2D and 3D plots, showcasing various dynamical structures along with diverse sets of parameters for a more comprehensive physical understanding. Additionally, we conduct a stability analysis of the governing equation, offering stability criteria along with the corresponding regions of stability. The outcomes are novel, valuable, and represent unexplored territory for the examined equation, as no prior studies have delved into these findings. [ABSTRACT FROM AUTHOR]

Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons interact elastically. However, soliton turbulence can also be a part of non-integrable dynamics, where long-lasting solutions in the form of almost solitons may exist. In the present paper, the complex dynamics of ensembles of solitary waves is studied within the Schamel equation using direct numerical simulations. Some important statistical characteristics (distribution functions, moments, etc.) are calculated numerically for unipolar and bipolar soliton gases. Comparison of results with integrable Korteweg–de Vries (KdV) and modified KdV (mKdV) models are given qualitatively. Our results agree well with the predictions of the KdV equation in the case of unipolar solitons. However, in the bipolar case, we observed a notable departure from the mKdV model, particularly in the behavior of kurtosis. The observed increase in kurtosis signifies the amplification of distribution function tails, which, in turn, corresponds to the presence of high-amplitude waves. • Schamel equation. • Soliton gas. • Soliton collisions. • Non-integrable equations. [ABSTRACT FROM AUTHOR]

The work studies some integrable properties and soliton type solutions of a non-autonomous Gardner equation with damping and forcing terms. A bilinear form, a bilinear Bäcklund transformation and a Lax pair are derived for the considered Gardner equation explicitly. K -Soliton solution with proper existence condition, smooth positons, breathers and their interaction solutions are presented via the bilinear form. Moreover, the amplitude as well as velocity of the soliton solutions are derived, and a first-order breather solution and a second-order smooth positon are generated from the two-soliton solution. The interaction between a single-breather solution and the single-soliton solution and the interaction of a second-order smooth positon and the single-soliton solution are studied analytically, based on the three-soliton solution. Profiles of various types of the obtained solutions and their interactions are illustrated graphically. • The integrability of a non-autonomous Gardner equation is presented through formation of bilinear form, a bilinear Backlund transformation and a Lax pair. • Soliton, smooth positons, breathers solutions and their interaction solutions are explored from non-autonomous Gardner equation via the Hirota's bilinear method. • Properties of wave solutions are studied through analytically and graphically. [ABSTRACT FROM AUTHOR]

This study aims to investigate the interactions of solitons with an external force within the framework of the Schamel equation, both asymptotically and numerically. By utilizing asymptotic expansions, we demonstrate that the soliton interaction can be approximated by a dynamical system that involves the soliton amplitude and its crest position. To solve the Schamel equation, we employ a pseudospectral method and compare the obtained results with those predicted by the asymptotic theory. The asymptotic theory predicts that the soliton interaction can be classified into three categories: (i) steady interaction occurs when the crest of the soliton and the crest of the external force are in phase, (ii) oscillatory behavior arises when the soliton speed and the external force speed are close to resonance, causing the soliton to bounce back and forth near its initial position, and (iii) non-reversible motion occurs when the soliton moves away from its initial position without changing its direction. However, the numerical results indicate the presence of an unstable spiral pattern. • Asymptotic methods for the Schamel equation. • First order ode system for the soliton crest position vs. soliton amplitude. • Soliton resonance with an external force. [ABSTRACT FROM AUTHOR]

Lie symmetry analysis of differential equations is an effective technique to compute exact solutions of a differential equation, to reduce number of independent variables or to reduce the order and nonlinearity of the equation. The present article focuses on the symmetry analysis of Vakhnenko–Parkes Equation (VP Equation). At first, Lie point symmetries of the VP Equation that permit an optimal system of one-dimensional subalgebras are discussed and then using the commutator table we construct the discrete symmetries of the VP Equation using the modified Hydon's method. Also, we compute the exact solutions using symmetry transformations. • We construct symmetries and first integrals of the Vakhnenko–Parkes equation. • Perform reductions. • Present graphs of the solitons. [ABSTRACT FROM AUTHOR]

The main attention of this paper focuses on the dynamical behavior and dispersive optical solitons in birefringent fibers with coupled Schrödinger-Hirota equation. Under the traveling wave transformations, the coupled Schrödinger-Hirota equation is reduced to plane dynamical system. With the help of the theory of planar dynamical system, we obtain a range of solutions which contain bell-shaped wave solutions, periodic wave solutions and kink-shaped wave solutions. Then by using the complete discriminant system method and symbolic computation, we give all the classification of single traveling wave solutions for the coupled nonlinear Schrödinger-Hirota equation. It is notable that the obtained results substantially improve or complement the corresponding conditions in the references [19, 20]. As a consequence, this paper gives a new idea to construct dispersive optical solitions for the coupled Schrödinger-Hirota equation. [ABSTRACT FROM AUTHOR]

Based on ansatz functions technique, we construct many types of novel wave structures and multiple lump-soliton solutions to the generalized Fitzhugh-Nagumo (gFN)-equation. In particular, we obtain entirely exciting lump, lump 1-kink, lump 2-kink, lump-periodic, manifold periodic-soliton, periodic cross-rational, kink cross-rational (KCR) solutions, interaction of the multi-lump and periodic solutions as well as breather style of solitary wave solutions. Using a transformation of dependent variable, which contain a controlling parameter (can control the direction, wave height and angle of the traveling wave), we build generalized breathers, rogue wave, Akhmediev breather, homoclinic breather, Ma breather, Kuznetsov-Ma breather and their relating rogue waves, multiwave, M -shaped rational and some various interactions. We show the results of our solutions graphically in distinct dimensions by assigning different values to the parameters. • The nonlinear physics for the fractional order perturbed Kaup-Newell model. • The analytical modified methods are applied. • Multiple solitons and multi-peak solitons with interactions [ABSTRACT FROM AUTHOR]

The present manuscript employed a rare approach to tackle a vector-coupled system of Chen-Lee-Liu Equation (CLLE). This approach was based on the combination of the Collective Variable Method (CVM) and the higher-order Runge-Kutta Method (RKM) to examine the dynamics of pulse parameters and total energy on the evolution of a system of CLLE. Furthermore, to expatiate more on the pulse propagation associated with the system, the resulting dynamical equations of motions were graphically illustrated with the help of Maple and Matlab mathematical software. Lastly, the present approach was not only proved to be efficient in pulse characterization, but was also believed to effectively tackle high-order coupled systems of complex evolution equations with more wave functions (more dependent variables). • A complete treatment of the coupled system of Chen-Lee-Liu equation is given. • Pulse characterization of the characteristic parameters is established. • A collective variables method alongside an efficient numerical method is employed. • Self-explanatory graphical illustrations are provided to support the established results. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the dynamical behavior, optical soliton solution and traveling wave solution for the fractional Biswas-Arshed equation with the beta time derivative by using theory of planar dynamical system. Firstly, by employing the traveling wave transformation and other integral transformations, the fractional Biswas-Arshed equation with the beta time derivative is simplified into two-dimensional planar dynamic system. Secondly, phase portraits for the fractional Biswas-Arshed equation with the beta time derivative are plotted. Finally, based on Professor Li's three-step method, optical soliton solution and traveling wave solution of the fractional Biswas-Arshed equation with the beta time derivative are constructed, the obtained solutions give the propagation of optical solitons in nonlinear optics. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani (KdVSKR) equation which consists of the KdV equation and the SK equation is studied. Soliton molecules of the KdVSKR equation are given by means of the velocity resonance mechanism. By selecting the values of the phases, soliton molecule bounded by the three solitons is transferred to other type of the soliton molecule bounded by the asymmetric soliton and one soliton. Multi-breather solutions are derived by selecting the complex conjugate relations in the parameters. The relative positions for the maximum amplitudes of the multi-breathers can adjust with different values of the phases. It demonstrates that the phases of the multi-soliton solutions play an important effect in certain phenomena. In the meanwhile, the interactions between a soliton molecule and one-order breather, and between a soliton molecule and one-order lump of the KdVSKR equation are analyzed. The interactions are an elastic collisions by both the analytical and graphical ways. [ABSTRACT FROM AUTHOR]

• A cubic–quartic complex Ginzburg–Landau equation is studied. • The enhanced Kudryashov's method is applied. • Bright and singular solitons are recovered for this model. This paper implements the enhanced Kudryashov's method to address cubic–quartic complex Ginzburg–Landau equation for locating its solitons. This is considered when chromatic dispersion is discarded because of its low count. Sis forms of self-phase modulation structures are studied and they are Kerr law, parabolic law, polynomial law, quadratic–cubic law, anti-cubic law and parabolic-nonlocal law. Thus, bright and singular solitons are recovered for this model. The existence criteria for such solitons have been indicated, as well. [ABSTRACT FROM AUTHOR]

• Fractional Tylor equation is introduced in this paper. • Our analysis shows that the main achievement in the mentioned paper is incorrect. • A counterexample is taken to verify our analysis. In this paper, some comments are presented for the paper by Ahmadian A. etc (Chaos, Solitons & Fractals 104 (2017) 68–76). The Caputo-type derivative is approximated by a linearization formula in the mentioned paper. However, our research shows that this approach is incorrect and some counterexamples are presented to verify our analysis. [ABSTRACT FROM AUTHOR]

• The N th-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique. • The higher-order soliton, first- and second-order breather solutions are constructed based on the N th-order Pfaffian solutions. • Hybrid solutions composed of breathers, lumps and solitons are derived. In fluid mechanics, the higher-dimensional and higher-order equations are constructed to describe the propagations of nonlinear waves. In this paper, we investigate the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. The N th-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique. The higher-order soliton, first- and second-order breather solutions are constructed based on the N th-order Pfaffian solutions. We graphically demonstrate that the amplitudes and velocities of the solitons are affected by some variable coefficients. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breathers, lumps and solitons included in the hybrid solutions possess the same properties as those of the breather and lump solutions. [ABSTRACT FROM AUTHOR]