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The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots. [ABSTRACT FROM AUTHOR]

In this work, we study the Chafee–Infante differential equation (CIDE) for various types of nonlinear waves (NW) like breather waves (BW), periodic cross-kink (PCKW), lump waves (LW), rogue waves (RW), periodic wave (PW), lump soliton (LS) with one kink, LS with two kink and periodic-cross lump waves (PCLW). The CIDE is a nonlinear evolution equation (NLEE) introduced by Nathaniel Chafee and Ettore Infante. This equation has been widely investigated in the context of pattern generation and soliton solutions, and it plays an essential role in understanding the complex nonlinear dynamics across a wide range of scientific fields. BW is a localized, non-dispersive solution to NW equations, maintaining shape and amplitude during travel. An LS is a type of solitary wave solution in NLEEs. In mathematics, RW refers to rare and unusually large solutions of NW equations. These waves have garnered significant attention due to their unpredictable and extreme nature. A PW is a type of wave that repeats its shape and behavior at regular intervals in both time and space. These waves are characterized by their periodicity, which means that they have a consistent and predictable pattern of oscillation. We'll offer a graphical explanation of our newly discovered answers toward the conclusion. [ABSTRACT FROM AUTHOR]

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we examine a novel generalized (2 + 1) -dimensional Bogoyavlensky–Konopelchenko (gBK) model analytically via application of five mathematical methods, namely called Extended Simple Equation (ESE) method, Modified Extended Auxiliary equation mapping (MEAEM) method, (G ′ / G) -expansion method, Exp (− Ψ (ϕ)) -expansion method and modified F-expansion method, respectively, to obtain different types solitary wave solutions in the form of trigonometric, hyperbolic, rational and exponential functions as compared to solutions exist in [S. T. Chen and W. X. Ma, Math. China 13 (2018) 525; A. Sonia, A. Jamshad, S.-U. Rehman and A. Ali, Int. J. Appl. Comput. Math. 9 (2023)]. The research focuses on physical systems, physiology, mathematical applications, engineering fields, and the chemical processes of species in the porous nanoparticles, examining the impact of various parameters on the nonlinear model. The graphs of some solutions are plotted in the form of two-dimensional and three-dimensional by assigning the particular values to the paraments with the assistance of mathematica software which illustrated the physical behavior of the concern model. The derived solution has numerous applications in fluid dynamics and the interaction of Riemann waves and long waves. [ABSTRACT FROM AUTHOR]

We investigate the rational solitons of the Kraenkel–Manna–Merle (KMM) system in ferrites via symbolic computation with the ansatz functions technique and logarithmic transformation. The KMM system describes the propagations of the ferromagnetic particles in ferrite materials. Multiwave solitons are reported by using the three-wave technique. Homocentric breathers (HBs) approach is also applied to build new forms of exact solutions. We construct M-shaped solitons and their dynamics are shown via the appropriate values of parameters. Two types of interactions for rational soliton with kink wave are investigated. Additionally, kink cross-rational solitons (KCRSs) and periodic cross-rational solitons (PCRSs) are examined via suitable transformation. [ABSTRACT FROM AUTHOR]

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of α which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail. [ABSTRACT FROM AUTHOR]

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of (2 + 1) -dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF's parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising. [ABSTRACT FROM AUTHOR]

In this paper, we address a newly nonlinear Schrödinger equation (NLSE) in (2 + 1) -dimensions influenced by cubic–quintic–septic law nonlinearity and spatial dissipations effects. We properly construct the dark and bright optical soliton solutions for our governing model. Furthermore, periodic, other singular solutions, multiwave, homoclinic breather, M-shaped rational and their interaction with kink waves of various structures will be derived. In addition to this, some 3D, 2D and contour profiles will be added to anticipate the wave dynamics. [ABSTRACT FROM AUTHOR]

In this paper, novel soliton solutions to the conformable time fractional Benjamin–Bona–Mahony and seventh-order time fractional Sawada–Kotera–Ito equations are investigated via application of two mathematical methods. These models are reformulated into a fractional order derivative by applying the conformable derivative operator. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic and exponential function solutions. For physical phenomena of concern models, some solutions are plotted in two-dimensional and three-dimensional by assigning the particular values to the parameters under the constrain conditions with the assistance of Mathematica software. Hence, all results obtained by our proposed methods are novel and have not been yet reported in any literature. Moreover, the proposed schemes are enormously superbly mathematical tools to review wave solutions of several fractional models in nonlinear science. [ABSTRACT FROM AUTHOR]

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two-kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave and multi-wave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two-kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present three-dimensional, two-dimensional, contour, and stream plot illustrations for our graphs. [ABSTRACT FROM AUTHOR]

In nonlinear Schrödinger equations (NLSEs), the third-order generalized NLSE is a significant sculpture which is utilized for modeling ultrashort pulses in fiber optics. In this study, we obtained wave and soliton solutions by using new extended and modified Rational expansion method to get several types of soliton such as bright solitons, dark solitons, perodic solitons and traveling waves. In three-dimensional and two-dimensional plots, we present graphical representations in dissimilar structures of some solutions to understand the phenomena physically. The development and achievements of computing show the power and effectiveness of current technology. In addition, we are able to resolve various other high-order NLSEs with the assistance of effortless and effectual technique. [ABSTRACT FROM AUTHOR]

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave, and multiwave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present 3D, 2D, contour, and stream plot illustrations for our graphs. [ABSTRACT FROM AUTHOR]

In this paper, under the observation of extended modified rational expansion method based on symbolic computation, the multiple solitary wave solutions for nonlinear two-dimensional Jaulent–Miodek Hierarchy (JMH) equation are constructed. In this investigation, we use the computer software Mathematica for the construction of multiple solitary wave solutions. The interested and important things in this work are the multiple solitary wave solutions which have various kinds of physical structures such as kink soliton, periodic traveling wave, bright soliton, anti-kink soliton, dark soliton, combined bright and dark solitons, topological soliton and peakon soliton. We are sure that the various kinds of soliton solutions are found first time by using one method in the existing literature works. On the basis of this research, we can say that the applied technique is very efficient, reliable, fruitful and powerful. The constructed soliton solutions for nonlinear JMH equation will play an important role in the investigation of different physical phenomena in nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this work, we establish novel solitary wave solutions of three nonlinear evolution dynamical models, namely, Boussinesq model, (3 + 1) -dimensional generalized shallow water wave (SWW) and (4 + 1) -dimensional Fokas dynamical models by applying two-variable (G ′ ∕ G , 1 ∕ G) - expansion technique. The SWW equations are usually appropriate when the fluid is in a state where the vertical length scale is significantly smaller than the length of horizontal scale. In computer simulations for ocean engineering, Boussinesq-type models are frequently used to represent water waves in harbors and shallow seas. In order to understand the physical processes of waves inside and on the surface of water, the Fokas dynamical model plays a crucial part in wave theory. Analytical solutions in different forms such as solitons, solitary waves, trigonometric, hyperbolic, rational function solutions, breathers-type waves and more wave solutions are devised through using the proposed method. The exact solutions are also presented in graphical form having applications in engineering and other areas of applied sciences. The obtained results show that the given technique is universal and efficient. In addition, this technique can also be applied on lots of other nonlinear dynamical wave models occurring in many scientific real-world application domains, including engineering sciences, mathematics physics, and many more. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system nonlinear (2 + 1)-dim Broer–Kaup–Kupershmit (BKK) equation under consideration is based on extension of modified rational expansion method. The various kinds of multiple soliton solutions named anti-kink soliton, traveling wave solutions, periodic solution, dark soliton, kink soliton, bright soliton, anti-kink bright solitons, kink bright solitons for nonlinear BKK system are successfully constructed under symbolic computation Mathematica. The calculated results are very interesting, different and novel which have not been investigated in past literature. The graphical demonstration of constructed solutions show by contour, 2D and 3D shape by using computer software Mathematica. The constructed multiple soliton solutions prove that EMRE approach is very straightforward, efficient, reliable and powerful for the investigation of other nonlinear partial differential equations. The investigated results will be described and play a keen role in the study of nonlinear physical phenomena in the area of physics and engineering. [ABSTRACT FROM AUTHOR]

Variational Integrator (VI) is a numerical technique, in which the Lagrangian of the system is used as the action integral. It is a special type of numerical solution that preserves the energy and momentum of the system. In this paper, we retrieve numerical solutions for heat and wave equation with the help of all possible combinations of finite difference scheme like forward–forward, forward–backward, forward–centered, backward–forward, backward–backward, backward–centered, centered–forward, centered–backward, centered–centered. We also use Lagrangian approach along with the projection technique to obtain approximate solutions of these linear models. This approach provides the best approximate solutions as well as preserves the energy of the system while the finite difference scheme gives only the numerical solutions. We also draw a comparison of existing exact solution with all approximate solutions for both models and provide graphical representation of these solutions. [ABSTRACT FROM AUTHOR]

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics with applications in optical fibers, plasma physics, and biomolecule dynamics. The focus of this paper is on four types of nonlinear Schrödinger equations, including the cubic nonlinear Schrödinger equation (CNLSE) and the Chen–Lee–Liu equation (CLLE). We present the existence of a Lagrangian and the invariant variational principle for two coupled equations of these types. Our investigation involves the approximation solutions of these equations, achieved by selecting a trial function with single or two nontrivial variational parameters in the rectangular box region in three cases. By using these trial functions, we found the functional integral and the Lagrangian of the system without loss. The general case for the two-box potential can be obtained based on a different ansatz. We approximate the Jost function by quadratic polynomials instead of a piece-wise linear function and then approximate it by the tanh function. Using the amplitude ansatz method, we derive the exact bright soliton, dark soliton, bright–dark solitary wave solutions, rational dark–bright solutions, and the periodic solitary wave solutions for the proposed equations. The solutions obtained from these techniques have a wide range of applications in various areas of physics and other applied sciences. The results will be presented in graphical representations including 2D, 3D, and contour plots, which highlight their effectiveness. These techniques can be utilized to solve numerous other nonlinear models that arise in mathematical physics and different other applied sciences fields. [ABSTRACT FROM AUTHOR]

Globally, infectious diseases pose a significant threat. Notable cases include influenza, Hepatitis B and HIV. COVID-19, caused by the coronavirus, has been the subject of recent discussion due to its great transmissibility. This study explores the Susceptible–Infectious–Recovered (SIR) epidemic model, explaining several analytical approaches to comprehend the nonlinear incident rates and geographic dispersion. The model investigates phenomena such as M-shaped solitons, homoclinic breather-like solitons, lump waves (LWs), lump solution (LS) with one or two kinks, rogue waves (RWs), periodic waves (PWs) and periodic cross-kink waves. These solutions aid in understanding the disease spread and informing containment strategies by identifying optimal outbreak control methods. This work studies localized wave solutions in nonlinear wave equations, known as soliton solutions including the LS. RWs are characterized by unexpectedly large amplitude and rapid profile changes, drawing attention for their erratic features. PWs exhibit periodic repetitions over time and space. Besides, some type of solitons with special waveforms are the periodic cross-kink, M-shaped and homoclinic breather-like solitons. Graphical representation visualizes the behavior of these effective waves, providing insights into virus spread in specific regions over time. SIR models help identify optimal strategies for controlling outbreaks. The work adds to our understanding of epidemic dynamics by illuminating how the movement of susceptible and infected individuals affects the spread of disease. [ABSTRACT FROM AUTHOR]

Ecological system is the interaction of biological community with other organisms and their physical environments. One of the important models is spatiotemporal prey–predator model of Michaelis–Menten-type functional response and reaction diffusion with constant harvest rate. This paper investigates the spatiotemporal structures of ratio-dependent prey and predator densities under different constraint conditions and suitable parameters. The ϕ 6 model expansion technique is followed to extract these densities. The prey and predator densities behave differently depending on the diffusion coefficients and other parametric ratios. The 3D and 2D behavior of these densities are also plotted and observed. These results may help for better understanding to study the biological community in a real marine environment or ecological system. [ABSTRACT FROM AUTHOR]

In this paper, we study the third-order nonlinear Schrödinger equation (NLSE) and optical metamaterial (OM) with quadratic-cubic nonlinearity (OM-QCNL) to describe the properties and presence of propagating solitary waves (SW) in a fibre optic medium. We present an effective field amplitude ansatz that allows us to obtain SW using the governing model. More specifically, in the presence of all orders of dispersion and various nonlinear variables, we find SW solutions of dipole, bright, dark and other forms. The graphical representation of these waves will also be presented. [ABSTRACT FROM AUTHOR]

This paper discusses the generalized (2 + 1) -dimensional Hietarinta-type fourth-order nonlinear equation (GHTE) and Resonant nonlinear Schrödinger equation (RNLSE). We will study the soliton solutions of the GHTE as well as RNLSE by the renowned, efficient and successful method called sub-ODE technique. The solutions are categorized as dark soliton, periodic, rational, positive, hyperbolic and Weierstrass elliptic solutions (WES). Stability analysis of the solutions is performed for both the models. Later on, these results will be displayed graphically by 3D, 2D, contour, density and streamline plots which describe the behavior of the waves. [ABSTRACT FROM AUTHOR]

In this paper, we have studied a generalized scale-invariant analog of the well-known Korteweg–de Vries (KdV) equation. The generalized scale-invariant analog of the Korteweg–de Vries (SIdV) plays as a bridge between the KdV equation. The generalized SIdV model was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing four mathematical methods, several types of exact and solitary wave solutions are established. For the physical behavior of the model, some solutions are plotted graphically by imparting specific values to the parameters under constrain condition. Hence, reconnoitered elucidations have profitable rewards in the field of mathematical physics. [ABSTRACT FROM AUTHOR]

This paper deals with the evaluation of multiwave, homoclinic breather, M-shaped rational solitons and interaction of M-shaped rational solitons with single and double kinks for (1 + 1) -dimensional longitudinal wave equation via the logarithmic transformation and symbolic computation with ansatz functions method. We will obtain two types of M-shaped rational solitons and their dynamics are revealed in figures by choosing the distinct values of involved parameters. In addition, two forms of interaction of M-shaped rational soliton with kink wave will also be evaluated. Furthermore, we will compute kink cross-rational solutions, periodic cross-rational solutions, generalized breathers and Akhmediev breathers for governing model. The solitons behaviors via different values of parameters in the produced solutions will be analyzed. [ABSTRACT FROM AUTHOR]

The Kaup–Newell equation is used to model sub-picoseconds pulses that travel throughout optical fibers. The fractional-order perturbed Kaup–Newell model, which represents extensive waves parallel to the field of magnetic, is examined. In this paper, two analytical techniques named, improved F-expansion and generalized exp (− ϕ (ζ)) -expansion techniques, are employed and new analytical solutions in generalized forms like bright solitons, dark solitons, multi-peak solitons, peakon solitons, periodic solitons and further wave results are assembled. These soliton solutions and other waves findings have important applications in applied sciences. The configurations of some solutions are shown in the form of graphs through assigning precise values to parameters, and their dynamics are described. The illustrated novel structures of some solutions also assist engineers and scientists in better grasping the physical phenomena of this fractional model. A comparison analysis has been given to explain the originality of the current findings compared to the previously achieved results. The results of computer simulations show that the procedures described are effective, simple, and efficient. [ABSTRACT FROM AUTHOR]

This template retrieves M-shaped rational solitons and their interactions with kink waves, homoclinic breathers, multiwave, Grey-black optical solitons, periodic cross-rational solitons and kink cross-rational solitons propagation in optical fibers where self-phase modulation (SPM) is negligibly minor and therefore removed. The proposed equation contains spatio-temporal dispersions (STD), of second and third orders, to recompense for small group velocity dispersion (GVD). All new analytical solutions are found by utilizing the symbolic computation with logarithmic transformation and ansatz functions approach. Moreover, stability characteristics of all solutions are found. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we have employed two different forms of mathematical methods, namely the extended simple equation method, and modified F -expansion method to establish several types of solutions of the generalized Kadomtsev–Petviashvili modified equal width-Burgers (G-KP-MEW-B) equation that is used to designate the propagation of long-wave with dissipation and dispersion in nonlinear media. A suitable transform is applied to convert into an ordinary differential equation. As a result, after implementation of the proposed schemes, distinct types of solutions are obtained in the form of exponential hyperbolic, trigonometric and rational functions. To analyze the physical phenomena of the model, some constructed solutions are plotted in 2-dimensional and 3-dimensional by inserting the specific values to attached parameters. Hence, the recommended schemes are highly admirably mathematical tools to evaluate the wave solutions of various models in nonlinear science. [ABSTRACT FROM AUTHOR]

This paper retrieves the investigation of rational solitons via symbolic computation with logarithmic transformation and ansatz functions approach for the (3 + 1) -dimensional generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt (GKDKK) equation in fluid mechanics, ocean dynamics and plasma physics. We find two categories of M-shaped rational solitons and their dynamics will be revealed through graphs by choosing the suitable values of involved parameters. In addition, two categories of M-shaped rational solitons and their interactions with kink waves will be analyzed. Furthermore, homoclinic breathers, multi-wave and kink cross rational solitons will be investigated. The periodic, rational, dark, bright, Weierstrass elliptic function and positive soliton solutions will also be retrieved with the aid of Sub-ODE approach. Moreover, stability characteristics of solutions will be evaluated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies the soliton solutions for Embedded soliton (ES) generating model with χ 2 nonlinear susceptibility. The bright, rational, Jacobi elliptic, periodic, dark, Weierstrass, hyperbolic solitary wave solutions will be found with the aid of sub-ODE technique under certain conditions. The main objective behind the sub-ODE method is to find the wave solutions of a complex model with the help of simple and solvable ODEs called sub-ODEs. The resulting wave solutions are presented graphically for suitable values of different parameters. [ABSTRACT FROM AUTHOR]

This paper retrieves different types of solutions such as M -shaped rational solutions with one and two kinks, homoclinic breather, multi-waves, periodic cross-kink and kink-cross rational solutions for Radhakrishnan–Kundu–Lakshmanan equation (RKLE). We will study the M -shaped interaction with rogue and kink, we will also obtain the M -shaped interaction with periodic and kink, periodic cross-rational solutions. At the end, we will express the graphical representation for our newly achieved solutions. [ABSTRACT FROM AUTHOR]

In this paper, the time incapable illimitable paraxial wave equation which is used in Kerr-media is being investigated. The paraxial wave equation has important applications in the light beam interaction within the ripple unit and also in calculation of the light emission outside the fluctuation. We have applied two techniques, namely, He's variational and Hirota bilinear techniques to extract the diversity of optical wave structures. The diversity of the wave solutions include the three-wave hypothesis, periodic cross kink, lump periodic, mixed type wave solutions and breather wave. The 3D, 2D, and their corresponding contour plots of the earned solutions are successfully drawn for the real, and imaginary behaviors by selecting the suitable values of parameters. [ABSTRACT FROM AUTHOR]

The principle goal of the paper is to present proficient limited contrast finite difference schemes to execute on the nonlinear coupled partial differential system which emulate the overseeing differential framework. In this paper, more consideration is given to the exactness and security of the proposed numerical schemes by review consistency and union of the arrangement which can be seen from figures and information tables. For the nonlinear differential system, mesh independent results are expensive which are accommodated by the generation of block tridiagonal matrix structures (inherent properties of schemes) which are measured in terms of L 2 & L ∞ norms which lead to a superb concurrence with the investigative arrangement. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multiple soliton interactions and other solitary wave solutions (SWS) for a perturbed nonlinear Schrödinger equation (NLSE) with negative index material having quadratic-cubic nonlinearity (NLSE-QCN). Due to its high order dispersion term, this model yields sub-picosecond impulses useful in mode-locked ring lasers. Hirota bilinear method (HBM) will be used to study soliton interaction. By controlling the parameters, we will obtain S , V , parabolic and anti-parabolic, butterfly, bright and dark shaped solitons. On the other hand, we will obtain some other solitary wave solutions with the help of Sine-Gordon expansion (SGE) scheme. [ABSTRACT FROM AUTHOR]

In this paper, the Allen–Cahn equation with time noise is under consideration. The extended fan-sub technique is used to find the exact solutions. The solutions are successfully extracted in the form of hyperbolic, trigonometric and mixed forms of solitons. Importantly, the physical unique value problems of the solutions are discussed using the different values of parameters. The 2D, 3D, and their corresponding contour behaviors of these solutions are depicted by choosing the different values of parameters. The stability is controlled through the Borel coefficient in the noise term. [ABSTRACT FROM AUTHOR]

In this paper, we will use various ansatz transformations to obtain different kinds of optical solitons solutions for couple of nonlinear Schrödinger equations (NLSEs). We get bright, dark, bright in the dark (dipole), combined dark–bright (combo) and some other solitary wave solutions for quadratic-cubic quintic system in optical meta-materials (CQS-OM) having quadratic cubic nonlinearity along with NLSE with anti-cubic nonlinearity (AC) in an optical fiber. We also show our results graphically in various dimensions under different constraint conditions. [ABSTRACT FROM AUTHOR]

Various forms of rational solitons, breathers, rogue wave, periodic wave and multiwave for Davydov solitons in α -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. [ABSTRACT FROM AUTHOR]

Analysis of simple analytical wave solutions to nonlinear evolution equations is a fascinating and growing area of knowledge in highly nonlinear sciences. The Riccati equation rational expansion technique is being used in the paper to establish the exact traveling wave solutions of the Phi-4 model. We have investigated certain types of explicit solutions using this technique, such as periodic, hyperbolic, and rational solutions with some free parameters. The solitary wave solutions are derived from the traveling wave solutions when the parameters are set to particular values. In mathematical physics, it is discovered that the implemented technique is more effective for finding the analytical wave solution of nonlinear evolution models. The approach stands out for its simple and quick computation procedure. [ABSTRACT FROM AUTHOR]

In this paper, we study the perturbed nonlinear Schrödinger equation (P-NLSE) representing the propagation of waves and may be regarded as a nonlinear complicated physical model. We use the modified extended Tan hyperbolic function (METhF) approach and the ( G ′ G 2 ) -expansion approach to construct some new traveling wave structures. Several solutions have been found including dark soliton, periodic-type solitons, bell-shaped solitons, single bell-shaped solitons. We also show a graphical representation of a number of exact solutions to the dynamical model together with a description of their behavior. The proposed techniques can also be extended to various nonlinear evolution models in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

In this work, we retrieve a series of new wave solutions of the double dispersive equation that describes the nonlinear wave propagation in the elastic inhomogeneous Murnaghan's rod by applying a novel integration norm known as the Sardar sub-equation method (SSEM). The solutions recovered by this method can be categorized as topological, non-topological, combo topological–non-topological, periodic and mixed periodic, singular and mixed singular solutions. In addition, by allotting different values to parameters, the physical movements of attained solutions are shown by plotting the 3D, 2D and contour graphs. On the basis of achieved results, we may claim that the proposed computational method is direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations, especially in engineering and applied sciences. [ABSTRACT FROM AUTHOR]

In this paper, we study chirp-free solitons (dromions) for the higher-order nonlinear Schrödinger equation (HNLSE) having non-Kerr terms with the aid of sub-ODE scheme. We find bright and singular dromion with domain walls, rational and periodic solitons, Weierstrass elliptic functions (WEFs), Jacobi elliptic solutions (JESs) and other solitary wave solutions for our governing model. The Sub-ODE scheme is very useful scheme, providing so many new solitary wave solutions especially JES and WEF solutions. We will also construct graphs of our results in different dimensions along with constraint conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study different analytical solutions for long water-wave equation (LWWE) like lump, breathers, rogue waves, periodic cross-kink, periodic wave, lump with one and two-kink, multiwave and periodic-cross lump. We compute the essential relationship between interaction between lump periodic and kink wave, periodic cross-kink and other types of exact solutions by studying the unknown nonlinear evolution equations (NLEEs) for long water wave. We show the graphical representation for our retrieve analytical solutions to observe the behavior of effective waves. [ABSTRACT FROM AUTHOR]

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on the Davey–Stewartson (DS)-II equation, and the extended modified auxiliary equation mapping (EMAEM) architectonic is used to develop a new set of solutions such as kink, singular kink, rational, combined soliton-like solutions, bell-type solutions, trigonometric and hyperbolic solutions. Furthermore, this study reveals that the used technique is efficient for solving other nonlinear evolution equations (NLEEs). The Painleve test (P -test) will also be used to examine the integrability of the DS-II equation. Finally, graphical simulations are designed to show the exact behavior of solutions as well as the efficacy of the suggested strategy. [ABSTRACT FROM AUTHOR]

This paper studies the exact traveling wave solutions to the nonlinear Dullin–Gottwald–Holm model which has the application in shallow-water waves in which the fractional derivative is considered in the sense of conformable derivative. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtained using two integration norms. For this purpose G ′ / G -expansion method and reccati mapping techniques are used. The 3D plots and their corresponding contour graphs are also depicted. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the modified Jacobi elliptic function method is applied for Salerno equation which describes the nonlinear discrete electrical lattice in the forbidden bandgaps. Dark and bright solitons are obtained. Also, periodic solutions and periodic Jacobi elliptic function solutions are reported. Moreover, for the physical illustration of the obtained solutions, three-dimensional and two-dimensional graphs are presented. [ABSTRACT FROM AUTHOR]

This paper studies Attilio Maccari (AM) system for some new solitary wave solutions. We used sub-ODE scheme to find kink shape, bell type, Weierstrass and Jacobi elliptic, hyperbolic, periodic and other solitary wave solutions under some constraint conditions. We will also present our results graphically in distinct dimensions. [ABSTRACT FROM AUTHOR]

In this research paper, a simple integration scheme is executed to secure new dark and singular soliton solutions for the highly dispersive nonlinear Schrödinger's equation having Kudryashov's arbitrary form with generalized nonlocal laws and sextic-power law refractive index. [ABSTRACT FROM AUTHOR]

Lump (rational) waves and their interactions with kink and periodic waves, periodic cross-lump solutions will be discussed for (2+1)-dimensional Maccari-system in this paper. With the combination of rational, exponential, and trigonometric functions, we will study various lump soliton solutions. We will find out analytical solutions with interaction phenomenon and also describe them in graphical ways. [ABSTRACT FROM AUTHOR]