Search

In this paper, the Riemann‐Hilbert (RH) approach for the integrable discrete Hirota equation with bounded boundary conditions is presented. In the direct scattering problem, we study the analyticity, asymptotics, symmetries of the eigenfunctions, and scattering coefficients and analyze the distribution of discrete eigenvalues. In the inverse scattering problem, the RH problem is constructed and solved as well as the reconstruction formula of potential is derived based on asymptotics. Finally, combining the time evolution, we solve the first‐ and second‐order dark soliton solutions on the nonzero background under the reflectionless condition. [ABSTRACT FROM AUTHOR]

In order to reduce the computational performance requirements and improve the nonlinear representation ability of neural networks and the accuracy of solving soliton solutions of partial differential equations, this paper utilizes a deep learning machine to design PINNs containing adaptive nonlinear combined activation functions. In particular, the physical information present in the mathematical physics model is incorporated into the neural network through the use of the neural network's generalized approximation capabilities. The PINNs with adaptive nonlinear combined activation functions are employed to obtain analytical solutions of partial differential equations, which is predicted by neural networks with high accuracy. By analyzing the spatio-temporal dynamics of the solutions of the (1 + 1)-dimensional Burgers equation, the (1 + 1)-dimensional Schrödinger equation, and the (2 + 1)-dimensional Navier–Stokes equation equations, it has been demonstrated that the proposed approach yields more accurate prediction results. Moreover, deep neural networks have been successfully employed in the solution of soliton problems. [ABSTRACT FROM AUTHOR]

This study comprehensively explores the (3 + 1) -dimensional Mikhailov–Novikov–Wang (ℕ) integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified () , Khater II ( hat.II) method, and He's variational iteration (ℍ ) method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation's pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics. [ABSTRACT FROM AUTHOR]

The focusing modified Korteweg-de Vries (mKdV) equation with multiple high-order poles under the nonzero boundary conditions is first investigated via developing a Riemann-Hilbert (RH) approach. We begin with the asymptotic property, symmetry and analyticity of the Jost solutions, and successfully construct the RH problem of the focusing mKdV equation. We solve the RH problem when 1/S11(k) has a single high-order pole and multiple high-order poles. Furthermore, we derive the soliton solutions of the focusing mKdV equation which corresponding with a single high-order pole and multiple high-order poles, respectively. Finally, the dynamics of one- and two-soliton solutions are graphically discussed. [ABSTRACT FROM AUTHOR]

In this study, the modified exponential approach is used to investigate the fractional order longitudinal wave equation in a magneto-elastic circular rod, which represents the nonlinear interplay between dispersion and longitudinal wave velocity depending on the rod's material and geometry. The time-fractional order is used in the standpoint of conformable derivative. Comprehensive and descriptive waveform solutions, including kink-shaped, bell-shaped, and bright bell-shaped solitons, are obtained. The produced soliton solutions are advanced, unique, developed, and crucial. To demonstrate the physical appearance of the obtained solutions, 2D, 3D, and contour graphs of some of the obtained solutions are plotted. The extracted solutions demonstrate the physical significance of the model and express the pressure and electrostatic potential for the longitudinal wave equation and can be used to elucidate more complicated nonlinear models of classical and fractional order. The obtained results are helpful in finding the amplitude of tsunami waves which is of great interest to researchers in the field of ocean engineering. If a tsunami extends across a large area, it becomes a manifestation of solitons. The comparison of the obtained solutions is also given in the form of remarks 1 and 2 which shows the novelty of our work. [ABSTRACT FROM AUTHOR]

In this study, we particularly address the generalized (3+1)-dimensional Kortewegde Vries (KdV) problem as one variation of the KdV equation. This equation can be utilized to simulate a wide range of physical events in a variety of domains, such as nonlinear optics, fluid dynamics, plasma physics, and other fields where coupled wave dynamics are significant. We first construct a Hirota bilinear form for the generalized KdV equation, and then we derive two different Bäcklund transformations (BT). The first Bäcklund transformation includes eleven arbitrary parameters, while the second form contains eight parameters. Rational and exponential traveling wave solutions with random wave numbers are found based on the suggested bilinear Bäcklund transformation. These solutions of the rational and exponential functions lead to the formation of dark and bright solitons. Moreover, we utilize the bilinear form of the equation to fully comprehend the behavior of lump-kink, breather, rogue, two-wave, three-wave, and multi-wave solutions. In-depth numerical simulations using 3-D profiles and contour plots are carried out while carefully taking into account relevant parameter values, offering more insights into the unique characteristics of the solutions that are obtained. Our results demonstrate the effectiveness and efficiency of the method used to obtain analytical solutions for nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The objective of this research is to investigate the initial value problem of a higher-order matrix-type nonlinear Schrödinger (NLS) equation with discrete spectrum as simple and double poles (simple and second-order zeros rank(P(x,t,kn))=3 and second- and third-order zeros under rank(P(x,t,kn))=2) under zero boundary conditions (ZBCs). Specifically, we not only consider the analyticity, symmetries and asymptotics of the Jost function, scattering and reflection coefficients in the process of direct scattering, but also residue conditions, norming constants, RH problem and the reconstruction formula in the process of inverse scattering. There exist some differences between it and the RH method in the study of vector and scalar equations, like the order of each pole of a−1(k) being less than or equal to the order of the zeros of deta(k) (assuming k=kn∈ℂ+ is a second- or third-order zero of deta(k) under rank(P(x,t,kn))=2(⇔rank(a(kn))=0)), etc. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive analysis of the (1 + 1) -dimensional Klein-Gordan equation which plays a significant role in various areas of theoretical and applied physics. The main focus of this research centers on several key areas. First, infinitesimal generators of symmetries were found by Lie symmetry invariance analysis. Then, using the adjoint representation, an ideal system was created based on the found Lie vectors. Secondly, by utilizing the analytical approach, namely the modified Sardar sub equation method, we systematically derive various novel soliton solution in the form of dark, bright, periodic, singular, combo, hyperbolic as well as mixed trigonometric. Finally, bifurcation analysis is performed at the system's fixed points, revealing chaotic behavior when an external force is introduced into the dynamic system. To identify the chaotic characteristics, a range of tools, such as 3D and 2D phase plots, time series, Lyapunov exponents, and multistability analysis, are utilized. Additionally, the sensitivity analysis of the model is examined under different initial conditions. These results enhance the understanding of nonlinear wave phenomena in mathematical physics and hold potential applications across numerous scientific disciplines. [ABSTRACT FROM AUTHOR]

Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs of matrix eigenproblems as a foundation, binary Darboux transformations are constructed. These transformations, initiated with a zero seed solution, facilitate the generation of soliton solutions for the SS-type integrable matrix mKdV equations presented. [ABSTRACT FROM AUTHOR]

This paper explores in detail the integrable Akbota equation, a Heisenberg ferromagnet-type problem that is essential to the study of surface and curve geometry. A variety of soliton families are represented by the generalized solitonic wave profiles that are produced using the improved modified Sardar sub-equation technique, which is renowned for its accuracy and dependability. There has never been a study that used this technique before the current one. As a result, the solitonic wave structures have kink, dark, brilliant, king-singular, dark-singular, dark-bright, exponential, trigonometric, and rational solitonic structures, among other characteristics. In order to check the energy conservation, the Hamiltonian function is created and energy level demonstrated. The sensitivity analysis is also presented at various initial conditions. The graphical representation is also depicted along with the appropriate parametric values. [ABSTRACT FROM AUTHOR]

In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely ( G ′ G ′ + G + A) -expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the work by L.A. Belkaryan and A.L. Belkaryan published in this journal (64 (7), 1472–1490 (2024)). A theorem is proved formulated as a conjecture in the preceding paper stating the existence and uniqueness of soliton solutions and corresponding solutions of the functional differential equation from a dual pair "function–operator." For the model of traffic flow on a Manhattan lattice, this result is used to study soliton solutions with more complicated characteristics than those specified by an additive cyclic subgroup of . [ABSTRACT FROM AUTHOR]

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry. [ABSTRACT FROM AUTHOR]

The inverse scattering transform is considered for the complex Sharma-Tasso-Olver equation with zero boundary condition by Riemann-Hilbert method. Under the reflection-less situation, we investigate the Riemann-Hilbert problem with one high-order pole and multiple high-order poles, respectively. By Laurent expansion of the Riemann-Hilbert problem and elimination of the integral factor involved in the solution, the explicit N-soliton solutions of the equation are derived. The interactions of several various solitons are displayed and their dynamics are analyzed. [ABSTRACT FROM AUTHOR]

The extended Gerdjikov–Ivanov equation in dense wavelength division multiplexed system for both kerr and parabolic law nonlinearities have been studied using the exp (- ϕ (ζ)) -expansion method. Completely new parameterized travel wave solutions are extracted. When special values are assigned to the parameters, Kink soliton solutions, periodic soliton solutions, and rational function solutions are derived. Figures have been added to indicate the behavior of solutions in some selected cases. [ABSTRACT FROM AUTHOR]

This study investigates the two-dimensional nonlinear complex coupled Maccari equations, which are significant in describing solitary waves concentrated in small spatial regions. These equations have applications across various fields, including hydrodynamics, nonlinear optics, and the study of sonic Langmuir solitons. Using the Bäcklund transformation, we explore a broad range of soliton solutions for this system, focusing on their spectral properties. The proposed method stands out for its simplicity and comprehensive results compared to traditional approaches. The obtained solutions are expressed in rigorous, trigonometric, and hyperbolic forms, providing deeper insights into the dynamics of the system. To enhance understanding, we present contour and three-dimensional graphical representations of the solutions. This study has potential applications in energy and industry by advancing the understanding of nonlinear wave phenomena, which are crucial in optimizing energy transfer processes and designing efficient systems in hydrodynamic and optical engineering. Additionally, the soliton solutions obtained here contribute to technologies in power transmission and high-speed optical communications, offering a foundation for innovations in sustainable energy systems and industrial applications. [ABSTRACT FROM AUTHOR]

In this paper, we analyze and provide innovative soliton solutions for a (2+1)-dimensional generalized Korteweg-de Vries (gKdV) problem. We obtain phase shifts and dispersion relations by using the generalized Arnous technique and the Riccati equation approach, thus allowing different soliton solutions to be developed. Several precise solutions with special structural properties, including kink and solitary soliton solutions, are included in our study. This detailed examination demonstrates the complex behavior of the model and its capability to explain a large scale of nonlinear wave occurrences in many physical settings. Thus, in scientific domains such as fluid mechanics, plasma physics, and wave propagation in media ranging from ocean surfaces to optical fibers, our results are crucial to comprehend the principles behind the production and propagation of many complicated phenomena. Finally, we provide 2D and 3D graphs for various solutions that have been obtained using Maple.

This paper was concerned with the inhomogeneous optical fiber model, which was governed by a nonlinear Schrödinger equation with variable coefficients. By spectral analysis for Lax pair of the equation, a corresponding Riemann-Hilbert problem was formulated. By solving the Riemann-Hilbert problem with simple poles, the formula of multi-soliton solutions was derived. Finally, we considered a soliton control system and obtained the one-soliton and two-soliton.

This examination analyzes the integrable dynamics of induced curves by utilizing the complex-coupled Kuralay system (CCKS). The significance of the coupled complex Kuralay equation lies in its role as an essential model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various essential domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical procedures are operated to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. This study uses the generalized Riccati equation mapping (GREM) method to search for analytical solutions. This method observes single and combined wave solutions in the shock, complex solitary shock, shock singular, and periodic singular forms. Rational solutions also emerged during the derivation. In addition to the analytical results, numerical simulations of the solutions are presented to enhance comprehension of the dynamic features of the solutions generated. The study's conclusions could provide insightful information about how to solve other nonlinear partial differential equations (NLPDEs). The soliton solutions found in this work provide valuable information on the complex nonlinear problem under investigation. These results provide a foundation for further investigation, making the solutions helpful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study's methodology is reliable, robust, effective, and applicable to various NLPDEs. The Maple software application is used to verify the correctness of all obtained solutions.

Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.

Soliton solutions play a crucial role in modeling stable phenomena across optical communications, fluid dynamics, and plasma physics, owing to their stability and persistence in solving nonlinear equations. This study centers on the extended Sakovich equation, emphasizing the importance of soliton solutions in predicting and controlling localized wave behaviors, which advances nonlinear dynamics and its various applications due to its integrable properties and flexible soliton characteristics. This equation is applicable across diverse fields such as fluid dynamics, nonlinear optics, and plasma physics, where it effectively models nonlinear wave phenomena, including solitons and shock waves. Additionally, it provides crucial insights into wave propagation in biological systems and acoustics, making it a valuable tool for analyzing complex wave dynamics. Additionally, we investigate bifurcation and modulation instability within this equation, employing the improved Sardar subequation method and the ℛ′ℛ,1ℛ\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{{ {\mathcal R} }^{^{\prime} }}{ {\mathcal R} },\frac{1}{ {\mathcal R} }\right) method to derive solitary wave solutions. These methods yield a diverse range of waveforms – hyperbolic, trigonometric, and rational functions – validated rigorously using Mathematica software for accuracy. Graphical representations vividly display various soliton patterns, such as singular, multi-singular, periodic singular, kink, anti-kink, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped, highlighting their effectiveness in revealing innovative solutions. Furthermore, a comparative analysis verified the novelty of our derived soliton solutions. This research significantly contributes to advancing soliton solutions for the Sakovich equation, promising diverse applications across scientific disciplines.

This study investigates the two-dimensional nonlinear complex coupled Maccari equations, which are significant in describing solitary waves concentrated in small spatial regions. These equations have applications across various fields, including hydrodynamics, nonlinear optics, and the study of sonic Langmuir solitons. Using the Bäcklund transformation, we explore a broad range of soliton solutions for this system, focusing on their spectral properties. The proposed method stands out for its simplicity and comprehensive results compared to traditional approaches. The obtained solutions are expressed in rigorous, trigonometric, and hyperbolic forms, providing deeper insights into the dynamics of the system. To enhance understanding, we present contour and three-dimensional graphical representations of the solutions. This study has potential applications in energy and industry by advancing the understanding of nonlinear wave phenomena, which are crucial in optimizing energy transfer processes and designing efficient systems in hydrodynamic and optical engineering. Additionally, the soliton solutions obtained here contribute to technologies in power transmission and high-speed optical communications, offering a foundation for innovations in sustainable energy systems and industrial applications.

Modified Korteweg-de Vries equations are used to describe certain phenomena in nonlinear optics, fluid dynamics, plasma physics, ocean physics and gas dynamics. In this paper, we investigate a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the small-amplitude surface waves in a strait or large channel of slowly-varying depth, width and non-vanishing vorticity. Via the truncated Painlevé expansion, Painlevé-type auto-Bäcklund transformations are obtained. Based on the Painlevé-type auto-Bäcklund transformations, periodic solutions and one-soliton solutions are derived. The soliton-like solutions are constructed via the modified Kudryashov method. We graphically show the kink-type solitons of the soliton solutions. Graphic analysis shows that the shapes, characteristic lines and velocities of the solitons are related to α 1 (t) and β (t) , while the soliton backgrounds and amplitudes just depend on γ (t) , in which α 1 (t) , β (t) and γ (t) are the dispersive, dissipative and line-damping coefficients of the equation we investigated, respectively, where t is the temporal variable. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of optical soliton solutions for the perturbed Fokas–Lenells equation with conformable derivative using the Kudryashov auxiliary equation method. The studied optical solutions include a class of categories, comprising dark, mixed dark-bright, multi bell-shaped, bell-shaped, and wave optical solutions. Furthermore, we analyzed the magnitude of the perturbed conformable Fokas–Lenells equation by investigating the impact of the conformable parameter and the effect of the time parameter on the novel optical solutions. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results obtained illustrate that the proposed method is robust, efficient, and readily applicable for constructing new solutions to a wide range of nonlinear fractional partial differential equations. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics and mathematical physics. [ABSTRACT FROM AUTHOR]

We present distinct types of analytical waveform solutions of the dispersive concatenation model which involves spatio-temporal dispersion, Hamiltonian perturbation terms and multiplicative white noise. The including high-order dispersion terms have important practical applications in nonlinear fiber optics, ocean waves and the Heisenberg spin chain. The obtained waveforms comprise solutions involving the Weierstrass function, periodic solutions, doubly periodic solutions, and soliton solutions, illustrating the potentially rich set of localized and periodic wave solutions of the studied model. It is found that the system support the existence of different kinds of soliton solutions including bright, dark, singular, dark-singular combo soliton, and bright-dark combo solitons. The physical meanings of these solutions and constraint conditions for their existence in the nonlinear medium are also determined. By choosing suitable values of the parameters, the obtained structures are also illustrated graphically. The results demonstrate that the phase component of the newly found wave solutions exhibit a nontrivial form containing the white noise. [ABSTRACT FROM AUTHOR]

This paper explores the modified Korteweg-de Vries equation with a time-dependent coefficient in (2+1) dimensions. We establish a robust theoretical framework by identifying the Lax pair and constructing infinitely many conservation laws. Using the strategy of covariance, we derive the Darboux transformation with quasideterminants, enabling the construction of multisoliton solutions. Additionally, we employ the Levenberg–Marquardt artificial neural network to train one- and two-soliton solutions for various time-variable coefficients. Our symbolic computations reveal that while the soliton amplitude remains unaffected by higher dimensions and time-dependent coefficients in the evolution equation, the soliton velocity changes during propagation across different planes. The accuracy of these findings is demonstrated through various graphical representations for different time-variable coefficients. This study advances the understanding of soliton dynamics in complex environments and introduces a novel intersection of neural network techniques with classical soliton theory. [ABSTRACT FROM AUTHOR]

The present study employs the improved F-expansion and modified exp (- Z (ς)) -expansion function methodologies to generate an enormous amount of novel wave solutions for the (2 + 1) -dimensional extended shallow water wave equation. This equation has widespread applications in many scientific and engineering domains, such as oceanography, hydraulic engineering, flood risk assessment, coastal engineering, tsunami modeling, environmental monitoring, and research on climate change. The improved F-expansion technique yields traveling wave solutions in the trigonometric and hyperbolic trigonometric formats while modified exp (- Z (ς)) -expansion function generates these solutions in the rational and linear forms in addition to trigonometric and hyperbolic trigonometric forms. In this regard, a wide range of solutions, which incorporate the kink pattern, Z-pattern, singular bell pattern or singular bright, anti-kink pattern, singular anti-bell pattern or singular dark, singular periodic pattern, singular pattern, singular complexiton pattern and plane pattern solitary wave solutions are generated via these two techniques. The physical significance of the solitons and singular solitons solutions that originated using these two analytical approaches are also discussed in this work. This work also discusses the stability analysis of the model. Applying the earlier described approach, provide a number of graphical representations, such as surface, 2D, and contour graphics, that illustrate the computational and fluctuating characteristics of the produced solutions. [ABSTRACT FROM AUTHOR]

This study investigates the (3 + 1)-dimensional extended Kairat-II model using Lie-Bäcklund symmetry (LBS) and the improved modified extended tanh-function approach (IMETFA). The bifurcation and sensitivity analyses are conducted to understand the stability and chaotic behavior of the model. Through graphical visualizations of phase diagrams, Lyapunov exponents, power spectra, fractal dimension and recurrence plots, the complex dynamics and stability characteristics of the model are elucidated. Additionally, the research focuses on the derivation of dark soliton and various combo soliton solutions of the Kairat-II model. The derived soliton solutions are graphically displayed in 3D and 2D plots. The obtained results are new and have never been reported in the literature for the considered equation. [ABSTRACT FROM AUTHOR]

Abstract In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in $$(1+1)$$ -dimensional, by employing the analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.

In this investigation, the analytical behavior of two prominent nonlinear wave equations, namely the doubly dispersive equation (DDE) and the Ablowitz-Kaup-Newell-Segur equation (AKNSE), have been scrutinized. Bifurcation analysis, sensitivity as well as chaotic phenomena are also performed for the earlier-mentioned dynamical systems. These analyzes have profuse applications in the field of electrical circuits and control systems, phase transitions in materials, climate patterns, chemical reaction networks, forecasting market trends, signal processing, and quantum mechanics. Using an advanced mathematical technique, the exact solutions of the mentioned two-wave equations with singular bell-shaped soliton, bell-shaped soliton, anti-bell-shaped soliton, singular soliton, and singular periodic soliton have been studied. The technique utilized in the study is dependable for solving complex nonlinear problems in various natural science and engineering disciplines employed by many researchers. This study identified ten general solutions and ten particular solutions for the two mentioned equations that are novel and precise wave solutions. The solutions obtained in our study are significant not only for understanding the mentioned field but also may be used in revealing other interesting phenomena, such as the analysis of seismology, the study of compulsive collapse, and the study of the material effects and inner construction of solids.

Abstract In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: $$\beta$$ β , and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark–bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.

This research explores some modernistic soliton solutions to the (3+1)-dimensional periodic potential the Gross–Pitaevskii equation with a truncated M-fractional derivative plays a significant role in Bose–Einstein condensation, which describes the dynamics of the condensate wave function. The obtained results include trigonometric, hyperbolic trigonometric and exponential function solutions. Three techniques named: the $ \exp_a $ function method, the Sardar sub-equation method, and the extended $ (G'/G) $-expansion approach are employed to achieve a variety of new solutions for the governing model. More comprehensive information about the dynamical representation of some of the solutions is being presented by visualizing the 2D, 3D and contour plots. This work reveals a number of new types of traveling-wave solutions, such as the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, and the singular soliton solutions. These novel solutions are not the same as those that were previously studied for this governing equation. The presented techniques demonstrate clarity, efficacy, and simplicity, revealing their relevance to diverse sets of dynamic and static nonlinear equations pertaining to evolutionary events in computational physics, in addition to other real-world applications and a wide range of study fields for addressing a variety of other nonlinear fractional models that hold significance in the fields of applied science and engineering.

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]

Based on a modified Snyder–Mitchell model in nonlocal photorefractive media, spatiotemporal optical solitonary solutions are obtained and the formation and transmission characteristics of spatial solitons under the control of 2D optical lattice are studied by using self-similarity algorithm. The simulations show that the lattice order, modulation intensity and beam width significantly affect the shape and stability of spatial soliton in photonic lattice. The results have important guiding significance for the experimental study of spatial solitons. [ABSTRACT FROM AUTHOR]

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. [ABSTRACT FROM AUTHOR]

We investigate the standard binary Darboux transformation (SBDT) for an -component sdC integrable system. For this, we construct the Darboux matrix using specific eigenvector solutions associated to the Lax pair, not only in the direct space but also in the adjoint space, resulting in the binary Darboux matrix. By the iterative application of the SBDT, we derive quasi-Grammian soliton solutions of the -component sdC integrable system. We also examine the Darboux transformation (DT) applied to matrix solutions of the sdC integrable system, expressing solutions using quasideterminants. Additionally, we thoroughly discuss the DT applied to scalar solutions of the system, expressing solutions as ratios of determinants. Furthermore, we investigate the SBDT and its application to obtaining quasi-Grammian multikink and multisoliton solutions for the -component sdC integrable system. Additionally, we demonstrate that quasi-Grammian solutions can be simplified to elementary solutions by reducing spectral parameters. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is an important theoretical model for ultra-long and ultra-fast transportation. Therefore, the situation of its normal propagation and collision has significant implications for practical applications. For example, the nonlinear Schrödinger models can be utilized in the optical, plasma and fluid dynamics fields. However, current researches mainly focus on solutions of one dimensional or two dimensional Schrödinger equations with constant coefficients. Although these solutions possess good properties, they are often too unilateral. In this paper, our research mainly focuses on solutions of a (2+1)-dimensional coupled variable-coefficient Schrödinger equation. Through an explicit transformation, the solutions of this equation in the form of one-soliton, two-soliton, three-soliton, and breather-like bright-dark solitons can be acquired by utilizing the developed Hirota bilinear method. Furthermore, we will explicate that the collisions among two-soliton, two-breather-like bright-dark soliton are elastic with the help of the asymptotic analysis method. At last, we will exhibit their collisions via illustrations. [ABSTRACT FROM AUTHOR]

In this study, we aim to obtain numerical results of the modified Benjamin-Bona-Mahony equation, Ostrovsky-Benjamin-Bona-Mahony equation, and Mikhailov-Novikov-Wang equation via the physics-informed neural networks (PINN) method. The equations are modeled for shallow and long water waves, as well as fundamental and phenomenal models in ocean engineering. According to the implementation, we obtained the PINN solutions of kink, bright, multi-soliton (two-soliton), and mixed dark-bright soliton solutions. According to the inference from the obtained results, we achieved good results in some cases compared to other approximate solution methods in the literature. However, it was also observed that the best possible results could not be obtained in cases where the soliton type was intricate and layered. While the results were obtained, the number of hidden layers and the number of neural networks in the layers also varied. These results are shown in tables. Since it is known that the aforementioned models are not solved by the PINN method, we anticipate that the study will lead to other studies in the field of ocean engineering. [ABSTRACT FROM AUTHOR]

The generalized mK(m,n) model is used to portray the movement of nonlinear waves by applying four mathematical methods with assistance of Mathematica software. These methods are called enhanced simple equation (SE) method, Exp(−Ψ(ξ))-expansion method, (G′∕G)-expansion method and modified F-expansion method. For physical behavior, few solutions are plotted graphically in the form of two-dimensional (2D) and three-dimensional (3D). The techniques and soliton solutions that are developed provide computational tools for additional research in this area. With this work, we have gained a better knowledge of soliton motion and how it is used in dynamical systems and other related technologies. Hence, this work will help physicists envision some new ideas and hypotheses in nonlinear science. [ABSTRACT FROM AUTHOR]

In this paper, we use the Lie symmetry analysis to obtain the closed-form solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili Benjamin–Bona–Mahony (gKP-BBM) equation. We obtain the infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for the gKP-BBM equation. Also, we evaluated the interactions of soliton solutions of the gKP-BBM equation. It is observed that multi-soliton complexes are self-localized in state wherein, several fundamental solitons including bright solitons and dark solitons are nonlinearly superimposed. A notable observation is the existence of topological defects in soliton interactions, where two solitons appear to be separated, with the adjoining structure being “out of phase” with each other. The kink, acting as a twist in the soliton’s value, overcomes topological defects, leading to a transition from one phase to another. This comprehensive analysis contributes to a deeper understanding of the dynamics and interactions of solitons in the (3+1)-dimensional gKP-BBM equation. [ABSTRACT FROM AUTHOR]