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Based on Darboux transformation and nonlinearization method, we construct rogue wave solutions of the higher-order nonlinear Schrödinger equation on two distinct Jacobi-periodic wave backgrounds, specifically, dnoidal and cnoidal waves. The distribution characteristics of the Lax spectrum for the Jacobi-periodic waves are shown at the λ complex plane. The effects of dispersion parameters δ 2 and δ 6 , the elliptic modulus k, and the integral constants C and C 1 on the dynamical behavior of rogue wave solutions under two different periodic wave backgrounds are systematically analyzed, and shown in detail through graphics. In particular, we find that the integral constant can achieve the controllable excitation of rogue waves at fixed points and the fission of higher-order rogue waves into rogue wave pairs, which exhibit some novel dynamical phenomena. [ABSTRACT FROM AUTHOR]

In this paper, a series of ripple waves with decay modes for the (3+1)$$ \left(3+1\right) $$‐dimensions Kadomtsev–Petviashvili equation, always viewed as nonintegrable KP equation, are investigated. The decay mode wave solutions including ripplon, lump ripples, and lump chain ripples are described by Airy function, and all solutions are constructed from the Gram determinant form. Their propagation dynamics behavior is studied, and a comprehensive analysis of the asymptotic properties of these solutions has been diligently conducted. [ABSTRACT FROM AUTHOR]

In this paper, our goal is to study the line-soliton solutions of the three-component Kadomtsev-Petrovashvili equation. Its solution also represents the interaction between the three fluid layers. Firstly, we obtain the analytical solutions of thex three-component KP equations, i.e., u 0 (x , y , t) , u 1 (x , y , t) , u 2 (x , y , t) , based on the KP theory and the three-component treatment of the τ -function. Secondly, we make the following conjectures about the solution structure of the line-soliton on the Grassmannian Gr(N, M). For u 0 , there are N bright soliton solutions when y ≫ 0 ; there are M - N bright soliton solutions when y ≪ 0 . For u 1 , there are N bright soliton solutions and N dark soliton solutions when y ≫ 0 ; there are M - N bright soliton solutions and M - N dark soliton solutions when y ≪ 0 . For u 2 , there are 2N bright soliton solutions and N dark soliton solutions when y ≫ 0 ; there are 2 (M - N) bright soliton solutions and M - N dark soliton solutions when y ≪ 0 . Again, we verified the accuracy of the conjecture by plotting the images of the soliton solutions on Gr(N, M). In addition, we analyse the rogue waves in the three-component KP equation and briefly explain why the maxima in the line soliton solutions arise. [ABSTRACT FROM AUTHOR]

Inhomogeneities in the wave field due to wave groups, currents, and shoaling among other ocean processes can affect the mean water level. In this work, the classical and unsolved problem of continuously computing the set-down and the following set-up induced by wave breaking on a shoal of constant finite slope is tackled. This is possible by using available theoretical knowledge on how to approximate the distribution of wave random phases in finite depth. Then, the non-homogeneous spectral analysis of the wave field allows the computation of the ensemble average by means of the phase distribution and the inversion of the integral of the second moment for the special case of a shoaling process with uniform phase distribution. In doing so, I am able to obtain a direct effect of the slope magnitude on the phases distribution. Therefore, an analytical and slope-dependent mean water level with continuity over the entire range of water depth is provided. [ABSTRACT FROM AUTHOR]

In this paper, a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation is systematically investigated based on the Hirota bilinear method. The explicit N-soliton solution and the bright and dark multi-soliton solutions of it are first derived. Next, various bright and dark higher-order breather solutions, including the periodic line wave solutions, as well as the hybrid solutions composed of solitons, breathers, and periodic line waves, are proposed by virtue of the complex conjugate constraints on the parameters. Then, applying the long wave limit to the N-soliton solution, the bright and dark lump solutions and line rogue wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation are constructed. The semi-rational solutions composed of breathers, lumps, solitons, and line rogue waves are further discussed. These new exact solutions all appear in pairs of bright and dark, which can be interpreted by the uplifts and collapses of energy. In addition, the dynamic behaviors of these exact nonlinear wave solutions are vividly demonstrated by their corresponding three-dimensional diagrams, sectional drawings, and density plots with contours. [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 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]

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the (3 + 1)-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields. [ABSTRACT FROM AUTHOR]

A theoretical investigation of nonplanar (cylindrical and spherical) amplitude modulation of dust acoustic waves (DAWs) in a four component dusty plasma including positive and negative dusts, Maxwellian ions and double spectral electron distribution (DSED) is carried out. The focusing modified nonlinear Schrödinger equation (MNLSE) is derived using the derivative expansion perturbation technique (DEPT). The limit of DSED to Maxwell distribution is discussed. The effects of the two spectral indices, $ r $ r and $ q $ q , on the modulational instability (MI) growth rate are examined and it is deduced that increasing the value of $ r $ r and $ q $ q raises the MI growth rate. Additionally, the structure of the first and second-order dust acoustic (DA) rogue waves in the modulational unstable region is analyzed. The influences of both of $ r $ r and $ q $ q on the rogue wave amplitude (energy) are investigated. We found that increasing the value of $ r $ r enhances the rogue wave amplitude while the opposite behavior is found by increasing $ q $ q values. The findings of the present study are useful to interpret some physical phenomena in various plasma situations that found in space regions, such as Titan (the largest moon) of Saturn, auroral region and asteroid zones, where the effectiveness of non-Maxwellian particles such as DSED is existed. [ABSTRACT FROM AUTHOR]

Under consideration is a new discrete coupled Ablowitz–Ladik equation with a time-dependent coefficient. Firstly, the continuum limit technique is used to correspond this discrete equation to two continuous variable coefficient equations. Secondly, the modulation instability on plane wave solutions is used to study the formation mechanism of diverse localized waves. Thirdly, the discrete generalized (m , N - m) -fold Darboux transformation is established and extended to solve this discrete equation. Finally, as an application of the resulting generalized Darboux transformation, abundant localized wave solutions including rogue wave, period wave and mixed interaction solutions are analyzed and presented graphically. Compared with its corresponding constant coefficient discrete equation, this variable coefficient equation has more abundant rogue wave structures, and it is found that a fundamental rogue wave can be split into two fundamental rogue waves through choosing time-dependent function and different control parameters. These new results and phenomena might be helpful for understanding the interaction and propagation of optical pulses. [ABSTRACT FROM AUTHOR]

Interactions of nonlinear waves play a significant role in physical systems. In this work, we attain the interactions of breathers and rogue wave for the coupled Lakshmanan–Porsezian–Daniel equation. Through the Darboux transformation, we succeed the interactional solutions consisting of different types of breathers and rogue wave. In particular, it can be observed that distinct parameters a 1 , a 2 , δ and λ 1 make the interaction properties, structures and energy conversion of breathers and dramatically change. Moreover, we exhibit the striking dynamics of interactional solutions based on three-dimensional figures. These results are helpful for the study of nonlinear waves in coupled integrable systems. [ABSTRACT FROM AUTHOR]

In this paper, using the Hirota bilinear method, we investigate the perturbed Gerdjikov–Ivanov equation (PGIE) for lump soliton, rogue wave, periodic and kink cross rational solutions, homoclinic breathers, M-shaped and many other interactions. All these solutions have so many real life applications for example lump wave help to study the behavior of vortices and eddies in the ocean, which are used for climate modeling and fisheries management. Rogue waves are used in maritime safety, M-shaped solitons are used in optical fiber communications to transmit data over long distances without distortion, improving the efficiency and reliability of high-speed data transmission. Breathers can be used in fiber optic systems to generate ultrafast optical pulses, which have applications in telecommunications and laser technology. We also discuss the dynamical behaviour of our solutions in various dimensions. [ABSTRACT FROM AUTHOR]

Under investigation in this work is a coupled nonlinear Schrödinger equation (CNLSE), which can be used to describe the dynamics of light beams and pulses in P T -symmetric coupled waveguides. Its breather wave (BW) and rogue wave (RW) solutions are presented here. The BW solutions can be converted into various soliton solutions including the Akhmediev breather, Kuznetsov–Ma soliton breather and Peregrine soliton. The dynamics of the solutions are graphically discussed. Moreover, we observe that the two BWs can evolve into an Akhmediev breather with a Peregrine soliton. We hope that the results can be used to enrich dynamical behavior of BWs and rogue waves in the CNLSE. [ABSTRACT FROM AUTHOR]

In this article, we mainly focus on investigating rogue waves in the Ito's system with the nonzero constant background and discussing their dynamic behaviors. Firstly, we discuss the linear instability of the nonzero constant background solution in the Ito's system and derive the instability condition. It is shown that the rogue wave only exists in the subset of the linearized instability region. Secondly, the Bäcklund transformation between two different solutions and linear representations of the Ito's system are derived, which provide convenience in solving the Ito's system. Finally, through the bilinear representation of the Ito's system and the parameter perturbation expansion method, we derive rogue wave solutions and discuss their dynamic properties. The findings show that the exponentially decayed rogue wave can be employed to describe the nonlinear evolution of the linear unstable mode solutions. Higher-order rogue waves with free parameters have more plentiful dynamic behaviors. The results show that the third-order rogue wave can not only exhibit triangle or pentagon patterns but also other novel hybrid patterns. These findings further enrich the patterns and dynamic features of rogue waves. [ABSTRACT FROM AUTHOR]

We probed the impact of both the degree of disorder and nonlinearity on rogue waves (RWs) in two-dimensional disordered lattices. Our results unveiled that an increase in the disorder level under linear conditions heightened the probability of RW occurrence and simultaneously contracted the "long tail". Interestingly, with the introduction of nonlinearity, this "long tail" became shorter compared with linear conditions. Nevertheless, in the context of disordered media, RW occurrence probability demonstrated relative stability—a distinct deviation from its conduct within homogeneous media. [ABSTRACT FROM AUTHOR]

In this communication, the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system is studied. With the aid of Painlevé Bäcklund transformation method, we have obtained a special class of solution consisting of only one arbitrary function. Consequently, we derive exact wave solutions, including dromion, dromion triplet pairs and multi-rogue waves. To realize dynamics of the wave patterns, we have created 3D plots with the use of symbolic software Mathematica. The obtained solutions are very significant to explore the qualitative interpretations, especially for the civil and coastal engineers in a coastal and harbor design. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the variable coefficient nonlocal Fokas–Lenells equation. On the basis of the Lax pair, the infinitely-many conservation laws and Nth-fold Darboux transformation are constructed. Depending on zero seed solution, soliton solutions are derived via the Darboux transformation. Based on nonzero seed solution, breather solutions and rogue wave solutions are obtained. The behaviors of solutions are clearly analyzed graphically. The influences of variable coefficient for solutions are discussed. The different profiles of solitons, breathers and rogue waves are observed via selecting different variable coefficients. Furthermore, the interaction of solitons and the interaction of breathers for the variable coefficient nonlocal Fokas–Lenells equation are both elastic. [ABSTRACT FROM AUTHOR]

This study considers an array of waveguides described by a discrete KdV equation. Rogue wave solutions are numerically derived for the dKdV equation under periodic and non-vanishing boundary conditions. When the dKdV equation is solved numerically under periodic boundary conditions, a discrete rogue wave occurs due to shock front breaking. Furthermore, the dKdV equation has been solved numerically under non-vanishing boundary conditions, and it has been found that the rogue wave amplitude depends on the ρ0 parameter. [ABSTRACT FROM AUTHOR]

We have obtained the first-order solution of a three-coupled nonlinear Schrödinger equation based on the modified Darboux transformation. In addition, we have derived an expression for the distance between the rogue wave and breather. We have find that the amplitude of the rogue wave, the period of the breather, the distance between the rogue wave and breather, the transformation between the bright and dark rogue wave, and the transformation between the soliton and breather which are all affected by the values of the free parameters. [ABSTRACT FROM AUTHOR]

The properties and stability of nonlinear dust-ion-acoustic waves are examined in dusty plasma environment such as Titan ionosphere. The plasma is composed of warm collisional negatively charged dust grains, positive ions contain source terms, and Maxwellian electrons. Based on the one-dimensional version of the standard reductive perturbation technique, the nonlinear waves are described by a complex Ginzburg-Landau (CGL) equation. Using a factorization method, the CGL equation is solved to obtain different nonlinear solutions in the form of a localized electrostatic potential pulse such as rogue, bright, and shock-like solutions. At certain plasma parameters, the rogue wave turns into a bright soliton or other nonlinear wave forms. The influence of various plasma configuration parameters, such as wave number k, relative temperatures σ e , coefficients of plasma ionization frequency H 2 , relative dust density δ d , and collision frequency ν on electrostatic nonlinear wave characteristics (amplitude and width) has been examined. The stability of the envelope soliton solution from the CGL equation has been investigated. Our results will be useful in understanding the dynamics of nonlinear excitations in Titan ionosphere where the dust grains have been well established in recent years by Cassini mission. [ABSTRACT FROM AUTHOR]

In this article, the equation showing the cold bosonic atoms in a zig-zag optical lattice model for some breathers, M-shaped solution and lump soliton solution, homoclinic breather pulses, breather lump pulses, periodic-cross kink wave, kink cross-rational propagation, and interaction between lump periodic and kink wave was examined. Some M-shaped solution, M-shaped interaction with periodic and kink, M-shaped interaction with rogue and kink, M-shaped rational solution, M-shaped rational solution with one kink, M-shaped rational solution with two kink, solutions for lump soliton waves, lump one kink waves, lump two kink waves, periodic-cross lump wave propagation, periodic wave propagation, rogue wave propagation, and multiwave propagation were also acquired. Likewise, our solution was also graphically presented, and also their stability was checked.

In this paper, based on the modified Darboux transformation, a new first-order solution of coupled fourth-order nonlinear Schrödinger equation (cNLS) is constructed. The amplitude of rogue wave, distance of the breather and the rogue wave can be changed if we adjust parameter d 1 . With the adjustment of the parameter c 2 , the breather and the rogue wave can be converted into each other, and the direction of propagation of the breather can be changed. When the initial wave height takes different values, images of the breather and the rogue wave as well as soliton-like and rogue waves can be presented. [ABSTRACT FROM AUTHOR]

By the q-to-Q relationship between the 3D coupled nonautonomous nonlinear Schrödinger system and Manakov model, with the help of the Darboux transform technique, vector fundamental bright–bright and bright–dark freak wave solutions with two different forms of coupled components are found. Analyzing the physical quantities of solution formation hints that diffraction parameters and phase chirps strongly affect the properties of freak waves including amplitude, width, center and phase. By selecting suitable parameters, we can realize three different cases to find the excitation controllability of vector bright–bright and bright–dark freak waves, including shape delayed excitation, shape maximum excitation and shape initial excitation. [ABSTRACT FROM AUTHOR]

This manuscript delves into the captivating domain of the (3+1)-dimensional Boussinesq equation, unraveling its complex dynamics and revealing mesmerizing soliton solutions. Our exploration begins with the application of the well-established traveling wave transformation, transforming the model into a set of ordinary differential equations (ODEs). Navigating through the intriguing intricacies of this system, we delve into bifurcations, chaos, and sensitivity to initial conditions, vividly brought to life through numerical simulations of phase portraits. Additionally, we present and discuss various solitary wave solutions, including singular, bright, breather-like, hybrid singular, rogue wave, and multi-singular periodic waves, vividly illustrating their existence through simulations. The (3+1)-dimensional Boussinesq equation presents rich dynamics with applications spanning fluid dynamics, climate science, optics, engineering, and more. Its soliton solutions and intricate behaviors contribute to advancements in diverse scientific disciplines.

Some new nonlinear wave solutions of the (3+1)-dimensional Ito equation, such as three-wave type solutions, lump wave solutions, rogue wave solutions and hybrid solutions of different forms, are studied by the bilinear neural network method. Besides, three-wave solutions are investigated via the single-layer neural network ‘4-3-1’ model, and hybrid soliton solutions between lump-type and strip-like soliton in ‘4-5-1’ model are discussed. Furthermore, rogue wave solutions are studied by the double-layer neural network the ‘4-2-3-1’ and the ‘4-3-3-1’ models, and the interference phenomena between them are analyzed using visual images. We also employed the extended parameter limit method to investigate the collapse degradation behavior of both solutions. Lump solutions and Y-type semi-rational solutions are derived separately.

We probed the impact of both the degree of disorder and nonlinearity on rogue waves (RWs) in two-dimensional disordered lattices. Our results unveiled that an increase in the disorder level under linear conditions heightened the probability of RW occurrence and simultaneously contracted the “long tail”. Interestingly, with the introduction of nonlinearity, this “long tail” became shorter compared with linear conditions. Nevertheless, in the context of disordered media, RW occurrence probability demonstrated relative stability—a distinct deviation from its conduct within homogeneous media.

The analytical rational solitons of the Kraenkel–Manna–Merle (KMM) system in a saturated ferromagnetic material will be explored in depth by using the symbolic computation with ansatz functions and logarithmic transformation. We investigate many systematic solutions such as lump soliton, rogue wave, lump with one kink, multiwave, periodic wave, periodic cross-kink wave and breather lump wave. We also examine M-shaped rational solution, homoclinic breather solution, M-shaped rational solution with one kink, kink-cross rational solution and M-shaped rational solution with two kink for KMM system. Additionally, we will draw graphs in different dimensions allowing analytical approaches to identify the structure of enormous waves. Furthermore, we look into the stability of a derived solutions and give a results in a tables. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional generalized shallow water wave equation is systematically investigated in this paper based on the Hirota bilinear method. The N-soliton solution and the higher-order kink-shaped breather solutions of the (3+1)-dimensional generalized shallow water wave equation are first proposed. Then, the line rogue wave solution and various hybrid solutions consisting of the breather, the kink-shaped soliton and the periodic solutions are discussed. Furthermore, the lump solutions of it are derived by using the long wave limit of the N-soliton solution. In addition, the diverse semi-rational solutions composed of lumps, kink solitons, line rogue wave and breathers enrich the research contents of the (3+1)-dimensional generalized shallow water wave equation. The dynamic behaviors of these exact solutions are vividly presented by their respective three-dimensional diagrams and density plots with contours. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) model is utilized to achieve the first- and second-order rogue wave solvers of the Boussinesq equation with different initial and boundary conditions. A new generalized loss term handling the initial first-order derivate is included in the simulations to guarantee the high prediction accuracies in the adaptive PINNs (APINNs) and the gradient-optimized PINNs (GPINNs) models, with a new regularization parameter being considered in the latter case. Learned results with high precision are fulfilled in the large domain simulations by applying more collocation points and more weight parameters in the neural network architecture. The APINNs model currently can be made use of in more situations with high prediction accuracies, while the GPINNs model is more robust in the current research where the initial condition is distributed in the localized sharp areas. Parallel computing is carried out to get the mean relative L 2 -norm errors efficiently in the GPINNs model due to the random choosing of the simulation points during the training iterations. [ABSTRACT FROM AUTHOR]

This article retrieves diverse forms of localized solutions including lump, lump one-strip, lump two-strip solutions for Schrödinger–Hirota equation with Kerr law nonlinearity through appropriate transformations technique. The proposed model applied in trans-continental and trans-oceanic distances in optical fibers. We compute lump solutions by choosing suitable polynomial function. Under various parameter framework, this lump solution have two forms of multiple-lump waves, particularly, one, and 2-lump waves. Mixed solutions with lump waves and solitons are also evaluated. In addition, we compute Akhmediev breather, generalized breathers and standard rogue wave solutions by using logarithmic transformation. Interaction behaviors are observed and we have also constitute few 3D and contour profiles for new solutions. Furthermore, we evaluate multi-wave soliton solutions for stated model via three wave transformation and logarithmic transformation. [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]

The Kuralay equation, as a Heisenberg ferromagnet-type equation, is of great significance for describing nonlinear phenomena in magnets. For the Kuralay-IIA equation, based on the constructed N -fold Darboux transformation, we derive its one-fold and two-fold rogue wave solutions, one-breathers and two-breathers including the Akhmediev two-breathers. By adding shifts on the time variable, we further separate the Akhmediev two-breathers into two one-breathers. In addition, the semi-rational solutions consisting of the rogue wave and one-breather solutions are also derived. The diagrams of the obtained solutions are plotted to analyze their dynamic features. [ABSTRACT FROM AUTHOR]

This article studies diverse forms of lump-type solutions for coupled nonlinear generalized Zakharov equations (CNL-GZEs) in plasma physics through an appropriate transformation approach and bilinear equations. By utilizing the positive quadratic assumption in the bilinear equation, the lump-type solutions are derived. Similarly, by employing a single exponential transformation in the bilinear equation, the lump one-soliton solutions are derived. Furthermore, by choosing the double exponential ansatz in the bilinear equation, the lump two-soliton solutions are found. Interaction behaviors are observed and we also establish a few new solutions in various dimensions (3D and contour). Furthermore, we compute rogue-wave solutions and lump periodic solutions by employing proper hyperbolic and trigonometric functions. [ABSTRACT FROM AUTHOR]

This paper investigates rational solutions of an extended Camassa-Holm-Kadomtsev-Petviashvili equation, which simulates dispersion's role in the development of patterns in a liquid drop, and describes left and right traveling waves like the Boussinesq equation. Through its bilinear form and symbolic computation, we derive some multiple order rational and generalized rational solutions and analyze their dynamic features, such as the connection between rational solution and bilinear equation, scatter behavior, moving path, and exact location of the soliton. The obtained solutions demonstrate two wave forms: multi-lump and multi-wave that consist of three, six and eight lump waves or two, three and four line waves. Moreover, different from the multi-wave solitons, stationary multiple dark waves are presented.

This work explores various wave pattern dynamics due to fractional derivative, dispersive and nonlinearity effects for the nonlinear time M-fractional modified Korteweg–de Vries (tM-fMKDV) model. To reconnoiter such dynamics, the unified and new form of modified Kudryashov’s techniques execute to integrate the nonlinear tM-fMKDV model for achieving diverse solitonic and travelling wave envelopes. As a result, trigonometric, hyperbolic and rational solutions have been found via a unified technique, and constants base wave solutions have been derived via the novel Kudryashov’s technique. The dynamical behaviors of the obtained solutions in the pattern of periodic waves, different types of periodic rogue waves, kink waves, and different types of double periodic waves have been illustrated with 3D and density plots for arbitrary choice of the permitted parameters. Analyzing the effect, anyone can observe that an increase in dispersion coefficient causes strictly shock to slopy shocked, an increase of nonlinearity reduces the wavelength and reduces fractionality causing smoothly banding wave shape to strictly banding. As a result, our findings demonstrate that the proposed schemes are highly effective, efficient, and accurate in capturing the characteristics of waves. Compared to other approaches, the solutions obtained from our tM-fMKDV models are more abundant.

Under consideration is the Heisenberg ferromagnetic equation, which can describe the motion of the magnetic vector of isotropic ferromagnetism. Starting from its 2 × 2 Lax pair, new location controllable rogue wave, period wave and different mixed localized wave interaction solutions on the sine-cosine function period background are obtained by the iterative generalized (m , N - m) -fold Darboux transformation method, especially we also give the movable dark RW with two peaks and one valley on the nonzero constant background and RW structure with two peaks and two valleys on the zero seed background. Compared with the known results, the main advantage of these new localized wave solutions is that they contain some position control parameters. By changing these parameters, we can theoretically control these localized waves to where we want them to appear. These new results may be useful for explaining and understanding the motion of the magnetic vector of isotropic ferromagnetism. [ABSTRACT FROM AUTHOR]

In this paper, the spatial discrete Hirota equation is investigated by Darboux–Bäcklund transformation. Firstly, the pseudopotential of the spatial discrete Hirota equation is proposed for the first time, from which a Darboux–Bäcklund transformation is constructed. Comparing it with the corresponding onefold Darboux transformation, we find that they are equivalent because there is no difference except for a constant times. We believe that this equivalence may hold universal if these two transformations are all derived from the same discrete spectral problem and using the similar technique in the references. Secondly, starting from vanishing and plane wave backgrounds, a variety of nonlinear wave solutions, including bell-shaped one-soliton, three types of breathers, W-shaped soliton, periodic solution and rogue wave are given, and the relevant dynamical properties and evolutions are illustrated by plotting figures. The relationship between parameters and solutions' structures is studied in detail, and the related method and technique can also be extended to other nonlinear integrable equations. Finally, we show that the continuum limit of breather and rogue wave solutions of the spatial discrete Hirota equation yields the counterparts of the Hirota equation. The results in this paper might be useful for understanding some physical phenomena in nonlinear optics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate various nonlinear phenomena of two Kadomtsev–Petviashvili (KP) equations. Based on bilinear neural network method, we present the fission and annihilation phenomena of breather waves and rogue waves on non-zero backgrounds of a (3+1)-dimensional KP equation by its reduced equation. Then with the aid of symmetry transformation and Hirota bilinear form, we obtain three interaction solutions on nonconstant backgrounds of a (2+1)-dimensional KP equation with variable coefficients. Also, we perform the analysis of the dynamic characteristics and evolution behaviors of the obtained solutions through three-dimensional animations with proper choices of different values for the parameters. This paper shows that the bilinear neural network method combined with symmetry analysis effectively solves high-dimensional differential equations with constant and variable coefficients. [ABSTRACT FROM AUTHOR]

The general rogue wave solutions for the one-dimensional (1D) Yajima–Oikawa (YO) system are derived through Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Different from the previous work, we improve the construction of the differential operators to save the complicated recursiveness and obtain the rogue wave solutions in a purely algebraic expression. Based on this simple expression, the new shape of the third-order rogue waves' arrangement is found. Moreover, three types of fundamental rogue waves and the rogue wave patterns from second to fifth order are graphically illustrated. In particular, there exist N - 1 ( 2 ≤ N ) polygonal configurations of Nth-order rogue waves for the 1D YO system, which is proven to be related to the Yablonskii–Vorob'ev polynomial hierarchy. [ABSTRACT FROM AUTHOR]

In the study of the evolution of Gaussian beam in saturated nonlinear media, it is found that the probability of optical rogue waves changes with the change of nonlinearity. The light intensity distribution on the exit surface of nonlinear medium can be characterized by scintillation index, and the change of rogue wave corresponds to the evolution of scintillation index. The rogue wave probability shows a complex trend with the evolution of nonlinearity. The Lyapunov exponent and power spectrum method are used to determine that the probability of rogue wave is chaotic with nonlinear evolution. [ABSTRACT FROM AUTHOR]

The partially nonlocal vector nonlinear Schrödinger equation governs the partially nonlocal nonlinear response of the self-phase and cross-phase modulations in optics and intra- and interspecies atomic nonlocal interactions in BECs. The influence of different diffractions in x- and y-direction on the excitation of partially nonlocal vector rogue waves is hardly studied. This paper aims to study this issue based on a variable-coefficient (3+1)-dimensional vector nonlinear Schrödinger equation with the partially nonlocal nonlinearity for different diffractions by reducing into a constant-coefficient vector nonlinear Schrödinger equation, whose solutions are used to construct analytical solutions of the variable-coefficient vector equation. Exploiting the nonrecursive Darboux method, partially nonlocal vector bright–dark rogue waves and bright–bright rogue wave pairs are revealed. Versatile excitations of these rogue waves are discussed in the sinusoidal and exponential diffraction systems by comparing the maximal value of the accumulated time with the value of the excited location for rogue wave. For the same value of β 1 or β 2 , with the add of the magnitude β 20 or β 10 of the initial diffraction in the y- or x-direction, the amplitude and widths of maximally or early excited figurations of bright–dark rogue waves both increase in the exponential system. These results in this paper deepen our comprehension of ultrashort wave phenomena found in the various fields of physics and engineering including optics, BECs and plasma. [ABSTRACT FROM AUTHOR]

The control of rogue waves in mid-infrared SC generated by pumped femtosecond pulses in chalcogenic optical fibers abnormal dispersion has been studied. And the control effect of the time delay between the seed light and the pump light on the rogue wave in the mid infrared supercontinuum is numerically studied. A seed is emitted into the photonic crystal fiber with an appropriate time delay to control the soliton trajectory. The results show that the appropriate negative time delay will increase the possibility of the formation of rogue waves in the mid-infrared supercontinuum, while the appropriate positive time delay will inhibit the formation of rogue waves. In addition, we also numerically study the influence of the peak power of the pump light on the formation of rogue waves. With the change of peak power, rogue waves will be promoted or suppressed. We believe that this study will provide new insights into the control of rogue waves. [ABSTRACT FROM AUTHOR]

In this paper, based on the combination of Hirota's bilinear method and long wave limit technique, we investigate rational and semi-rational solutions to the third-type Davey–Stewartson (DS III) equation and its nonlocal version. Rational solutions to the DS III equation demonstrate to be kinks, lumps and line rogue waves, while semi-rational solutions display hybrids of solitons, lumps and line rogue waves. As to the nonlocal DS III equation, we derive (semi-)rational solutions and breather solutions. Semi-rational solutions show lumps on the periodic line backgrounds, hybrids of breathers and lumps, line rogue waves and line breathers on the periodic line background. [ABSTRACT FROM AUTHOR]

In this work, we investigate a generalized Kadomtsev–Petviashvili equation with variable coefficients and self-consistent sources in plasma and fluid mechanics. The multiple rogue wave solutions, including 1-, 3-, and 6-order rogue waves, are presented by three different functions under a nonlinear transformation. Based on the Hirota bilinear method and a more complex assumption, new lump solutions are constructed, which have not been seen in other literature. The dynamic properties of the obtained results are illustrated graphically.

In this paper, a coupled mixed derivative nonlinear Schrödinger system, which describes the short pulses in the femtosecond or picosecond regime of a birefringent optical fiber, is investigated. Based on the known Nth-order breather solutions, we derive the first-order breathers and investigate their properties, e.g., velocities and peak amplitudes, where N is a positive integer. Then, two kinds of the second-order breathers are presented. We construct the Nth-order semirational solutions, with only one spectral parameter involved. Based on the obtained Nth-order semirational solutions, we analytically investigate and graphically illustrate the vector degenerate breathers, rogue and breather-rogue waves. We discuss how certain parameters, e.g., the nonlinear coefficients, affect the shapes of the degenerate breathers, rogue and breather-rogue waves. [ABSTRACT FROM AUTHOR]