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The analytical study of Boiti-Leon-Manna-Pempinelli (BLMP) equation is presented in this research paper. In this study, two exact methods are utilized to attain the exact solution of proposed equation. The generalized projective Riccati equations method and modified auxiliary equation method are simple and effective techniques, which have been used to attain the exact soliton solutions of BLMP equation. Some novel exact solution of BLMP equation are acquired using of proposed methods. The obtained solutions contain rational, geometric, hyperbolic functions. The graphical simulations of attained solutions are represented by plotted graphs. The plotted graphs show different solitons patterns such as kink solitons, anti-kink soliton, dark singular soliton, bright singular soliton, dark-bright singular solition and some other singular solitons. Mathematical modeling, analysis of physical phenomena and dynamical processes can yield solutions that enhance our understanding of their dynamics, which can be leveraged to gain valuable insights into the behavior and characteristics of these systems. [ABSTRACT FROM AUTHOR]

The nonlinear space–time fractional Sasa–Satsuma and Schrödinger–Hirota equations with beta derivative describe optical soliton, photonics, plasmas, neutral scalar masons, and long-surface gravitational waves in the real world. Through the fractional wave transform, the models are converted into a single wave variable equation. In this article, we examine a range of compatible, useful, and typical wave solutions expressed in the forms of hyperbolic, trigonometric, and rational functions uniformly through the ( Q ′ / Q , 1 / Q )-expansion approach. When specific parameter values are set, the generalized wave solutions exhibit a wide range of shapes, including asymptotic, anti-asymptotic, dark-optical, breather, lump-periodic, kink, kink-bell-shaped, homoclinic-breather, bright, dark, and periodic solitons that resemble periodic breathing patterns. We also investigate the effect of the fractional parameter δ into the wave profile, revealing a clear correlation between changes in the fractional order derivative δ and variation in the soliton's shape. The results underscore the use of this approach for the exploration of diverse nonlinear fractional systems within the context of beta derivatives. Varying the fractional-order δ and maintaining specific fixed parameter values, we depict 3D-surface, 2D-surface, density, and contour plots to visualize some of the derived solutions. [ABSTRACT FROM AUTHOR]

This paper investigates a novel method for exploring soliton behavior in ion transport across biological membranes. This study uses the Hirota bilinear transformation technique together with the Poisson–Nernst–Planck equation. A thorough grasp of ion transport dynamics is crucial in many different scientific fields since biological membranes are important in controlling the movement of ions within cells. By extending the standard equation, the suggested methodology offers a more thorough framework for examining ion transport processes. We examine a variety of ion-acoustic wave structures using the Hirota bilinear transformation technique. The different forms of solitons are obtained including breather waves, lump waves, mixed-type waves, periodic cross-kink waves, M-shaped rational waves, M-shaped rational wave solutions with one kink, and M-shaped rational waves with two kinks. It is evident from these numerous wave shapes that ion transport inside biological membranes is highly relevant, and they provide important insights that may have an impact on various scientific disciplines, medication development, and other areas. This extensive approach helps scholars dig deeper into the complexity of ion transport, illuminating the complicated mechanisms driving this essential biological function. Additionally, to show the physical interpretations of these solutions we construct the 3D and their corresponding contour plots by choosing the different values of constants. So, these solutions give us the better physical behaviors. [ABSTRACT FROM AUTHOR]

The space–time fractional Biswas–Arshed and Schrödinger Kerr law equations featuring beta derivative hold substantial application in nonlinear optics, optical solitons, ultrafast optical signal, nonlinear photonics, quantum optics, biophotonics, photonic crystals photonics, etc. In this study, a wide variety of geometric shape solitons have been established that include hyperbolic, exponential, trigonometric, and rational functions, as well as their assimilation to the considered equations, through the two-variable ( R ′ / R , 1 / R )-expansion approach. The implication of the fractional parameter μ on the wave shape has also been examined by depicting two-dimensional and three-dimensional plots for particular parameter values. The solitons include irregular periodic, pulse like, V-shaped, bell-shaped, positive periodic, asymptotic, general solitons, and some others. It is significant to note that the changes in the wave pattern result from the adjustments to substantive and auxiliary parameters. The outcomes demonstrate the efficiency, acceptability, and dependability of the ( R ′ / R , 1 / R )-expansion approach for obtaining solutions to the fractional-order evolution equations in the domains of engineering, technology, and sciences. It is evident from the graph that changing the value of μ results in a change in the shape of the soliton. The study explores how these equations change as fractional-order derivatives vary. Soliton solutions, which are stable, localized waveforms, are crucial in optical communication systems. Understanding their behavior under changing fractional-order derivatives is essential for advancing optical signal processing and communication technologies. [ABSTRACT FROM AUTHOR]

In this paper, we use new extended generalized Kudryashov and improved tan (ϑ 2) expansion approaches to investigate Kerr law nonlinearity in the extended Gerdjikov–Ivanov equation in a dense wavelength division multiplexed system. These methods rely on a traveling wave transformation and an auxiliary equation. These approaches successfully extract trigonometric, rational and hyperbolic solutions, along with some appropriate conditions imposed on parameters. To explain the dynamics of soliton profiles, a graphical description of newly discovered solutions is also presented, which exhibits distinct physical significance. The considered methods are recognized as useful and influential tools for creating solitary wave solutions to nonlinear problems in the mathematical sciences. [ABSTRACT FROM AUTHOR]

In this manuscript, a new form of the generalized q-deformed Sinh-Gordon equation is investigated which could model physical systems with broken symmetries and to incorporate phenomena involving amplification or dissipation. The proposed model is explored based on the Lie symmetry approach. Using similarity reduction, the partial differential equation is transformed into an ordinary differential equation. By employing the generalized auxiliary equation approach, precise results for the derived equation are obtained. The solutions are graphically depicted as 3D, 2D, and contour plots. Furthermore, the qualitative analysis of the considered model is investigated by employing the concepts of bifurcation and chaos. The phase profiles are displayed for different sets of the parameters. Additionally, by applying an external periodic strength, quasi-periodic and chaotic behaviors are documented. Various tools for detecting chaos are discussed, including 3D and 2D phase patterns, time series, and Poincaré maps. Additionally, a sensitivity analysis is conducted for various initial conditions. The obtained findings are unique and indicate the viability and efficacy of the suggested strategies for evaluating soliton solutions and phase illustrations for various nonlinear models. [ABSTRACT FROM AUTHOR]

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the MRLW equation by a classic finite difference formula, while the space derivatives are approximated by a θ — weighted scheme. For comparison purposes, we also find a soliton solution using the Adomian decomposition method (ADM). The Sinc-collocation method was were found to be more accurate and efficient than the ADM schemes. Furthermore, we show that the number of solitons generated can be approximated using the Maxwellian initial condition. The proposed methods' results, analytical solutions, and numerical methods are compared. Finally, a variety of graphical representations for the obtained solutions makes the dynamics of the MRLW equation visible and provides the mathematical foundation for physical and engineering applications. [ABSTRACT FROM AUTHOR]

KdV equations have a lot of applications of in fluid mechanics. The exact solutions of the KdV equations play a vital role in the wave dynamics of fluids. In this paper, some new exact solutions of a generalized geophysical KdV equation are computed with the aid of tanh-coth method. To implement the tanh-coth procedure, we first convert the PDEs to ODEs with the help of wave transformation. Then, using a system of algebraic equations, we obtain several soliton solutions. To verify and clearly illustrate the exact solutions, several graphic presentations are developed by giving the parameter values, which are then thoroughly discussed in the relevant components. [ABSTRACT FROM AUTHOR]

The Dbar-dressing method is extended to investigate the derivative non-linear Schrödinger equation with non-zero boundary conditions (DNLSENBC). Based on a meromorphic complex function outside an annulus with center 0, a local Dbar-problem inside the annulus is constructed. By use of the asymptotic expansion at infinity and zero, the spatial and temporal spectral problems of DNLSENBC are worked out. Thus, the relation between the potential of DNLSENBC with the solution of the Dbar-problem is established. Further, symmetry conditions and a special spectral distribution matrix are presented to construct the explicit solutions of DNLSENBC. In addition, the explicit expressions of the soliton solution, the breather solution and the solution of the interaction between solitons and breathers are given. [ABSTRACT FROM AUTHOR]

This study investigates the Hamiltonian amplitude equation by using the generalized exponential rational function method. As a result, a variety of soliton solutions have been established with different wave structures such as optical exponential, dark, singular, periodic, periodic-singular, and hyperbolic solutions. 2D and 3D graphics of certain acquired solutions are drawn with the help of computer program. By picking appropriate values for the free parameters in the obtained solutions, the physical phenomena for these obtained solutions are graphically studied and depicted. [ABSTRACT FROM AUTHOR]

In this study, we tackle with the (2+1)-dimensional complex nonlinear two-coupled Maccari equation which is a kind of Schrodinger equation and presented by Attilio Maccari in 1996. The Maccari equations has an importance to define the motion of isolated waves concentrated in a tiny region of space in a variety of fields, including hydrodynamics, plasma physics, quantum field theory, nonlinear optics, the sonic Langmuir solitons and so on. We utilized the popular method namely new Kudryashov scheme, we obtained the soliton solutions, graphical representation and made its physical interpretation. [ABSTRACT FROM AUTHOR]

We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transformations of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots, and density profiles the evolution of solitary waves. [ABSTRACT FROM AUTHOR]

This paper presents new optical solitons of a fractal Fokas–Lenells equation that models the dynamics of optical fibers. The Painlevé technique is employed to identify kink soliton solutions. The constraint conditions guarantee the existence of these solitons. The outcomes of this research give new solutions that are not achieved using some already defined algorithms. The derived method is efficient and its applications are promising for other nonlinear problems. The 3D graphical illustrations of obtained results are depicted for various values of the fractal parameter. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Hirota–Satsuma coupled KdV equations that can be used to describe the interaction of two classes of long waves. By using the Hirota bilinear method, the 1, 2, 3-soliton solutions are obtained. On this basis, the asymptotic analysis of soliton solutions proves that the collisions between solitons are elastic, and a set of visual figure is given to illustrate the results. • Integral terms and mixed derivative terms are added to the equation to describe more complex practical problems. • Function expansions can get richer results with more general expansions. • Not only bright soliton solutions and dark soliton solutions, but also periodic soliton solutions are obtained. • A wealth of graphic material is provided for readers to understand. • The asymptotic behavior of 2-, 3- soliton solutions is studied. [ABSTRACT FROM AUTHOR]

Based on the bifurcation theory, a new (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation is studied. The phase portraits of this equation are drawn, and some exact solutions, including solitons and periodic wave solutions, are derived. The (G ′ / G) -expansion method is applied to obtain many exact solutions, such as bright soliton, singular wave, and periodic wave solutions. Moreover, the interesting physical structures of these obtained solutions are described. • A new (3+1)-dimensional Kadomtsev-Petviashvili equation is investigated. • The phase portraits of this equation by bifurcation theory are drawn. • The model is solved using the (G'/G)-expansion method. • Some soliton, singular and periodic wave solutions are derived. [ABSTRACT FROM AUTHOR]

In this work, we obtain one-fold and N -fold Darboux transformation for the integrable (2 + 1)-dimensional Fokas–Lenells equations by determinant representations. The local and Ablowitz–Musslimani type nonlocal reductions are presented to deduce new integrable systems. A key point for reduced systems is that the special eigenfunctions of spectral problem are used to guarantee the validity of the reduction conditions. Different from the nonlocal (2 + 1)-dimensional Fokas–Lenells equation, the relation between spectral parameters λ 2 j and λ 2 j − 1 is required in the study of Darboux transformation for local (2 + 1)-dimensional Fokas–Lenells equation. In view of reduction formulas and different zeed solutions, multi-soliton solutions are derived. We also illustrate one-soliton and two-soliton solutions by plotting their graphs for particular values of the parameters, some of which include bright solitons, periodic waves, shock waves, breathers, dark solitons, antidark solitons, interactions and parallel propagations of mentioned type of waves. Consequently, it is clearly shown that the solutions of nonlocal (2 + 1)-dimensional Fokas–Lenells equation have new characteristics which differ from the ones of local case. [ABSTRACT FROM AUTHOR]

This paper aims to deduce optical soliton solutions of the highly dispersive optical solitons with perturbed complex-Ginzburg–Landau equation having parabolic law nonlinear refractive index in birefringent fibers. The addendum Kudryashov's method and algebraic sub-equation expansion method are used for solving the given system. Combo soliton solutions, bright soliton solutions and singular soliton solutions as well as rational and periodic wave solutions have been deduced. [ABSTRACT FROM AUTHOR]

This paper aims to constructing the Darboux transformation and soliton solutions of the focusing and defocusing spatial discrete coupled complex short pulse (sd-CCSP) equation. With the Darboux transformation, we derived three types of soliton solutions for the focusing sd-CCSP equation, which include smooth-, cuspon- and loop-type. These solutions converge to exact solutions for the focusing CCSP equation in the continuum limit. The interaction of two solitons and their asymptotic behavior are analyzed. We also obtain three types of soliton solutions for the defocusing sd-CCSP equation, as well as the rational solutions for the focusing sd-CCSP equation. We find that there exist new properties of soliton solutions for the sd-CCSP equation which do not exist in the sd-CSP equation, and there is a big difference between sd-CCSP equation and spatial discrete coupled nonlinear Schrödinger (sd-CNLS) equation. • We construct the Darboux transformation for the spatial discrete coupled complex short pulse (sd-CCSP) equation. • We derive soliton, rational and rational-soliton solutions for the sd-CCSP equation. • The interaction of two solitons and their asymptotic behavior for the sd-CCSP equation are analyzed. [ABSTRACT FROM AUTHOR]

• All possible integrable shifted nonlocal NLS and MKdV equations are given. • One- and two-soliton solutions of these equations are obtained by Hirota method. • One-soliton solutions of shifted and standard nonlocal equations are compared. • The relations between solutions of shifted and standard nonlocal equations are discussed. We find one- and two-soliton solutions of shifted nonlocal NLS and MKdV equations. We discuss the singular structures of these soliton solutions and present some of the graphs of them. [ABSTRACT FROM AUTHOR]

In this paper, using Lie symmetry method, some new soliton and doubly periodic solutions of the complex Ginzburg–Landau equation (CGLE) with two forms of nonlinearity are obtained. The obtained solutions are implicit solutions and explicit solutions in the form of Jacobi and Weierstrass elliptic functions. Under some conditions, we can retrieve kink wave, dark and bright soliton solutions for the CGLE from the obtained doubly periodic solutions. [ABSTRACT FROM AUTHOR]