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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]

We examine multiwave (MW), rogue wave (RW), periodic wave (PW), homoclinic breather (HB), breather lump wave (BLW), M -shaped rational solutions, lump soliton, kink cross-rational (KCR), periodic cross-kink rational (PCKR), periodic cross-rational (PCR) solutions for the Degasperis–Procesi (DP) equation in shallow water waves through appropriate polynomial function scheme. We also compute some interactions for stated model including interaction of M -shaped soliton with one kink, interactional solution with two kinks also lump soliton with kink one and kink two solutions. Furthermore, we made suitable 3D, 2D and contour profiles via appropriate parameters. [ABSTRACT FROM AUTHOR]

Dynamical properties of vector localized and periodic waves hold significant importance in the study of physical systems. In this work, we investigate the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation. For the two interacting components, we construct the binary Darboux transformation formulas, vector localized, and periodic wave solutions. Via those solutions, different kinds of nonlinear waves can be achieved, including rogue waves, solitons, positons, and periodic waves. When the imaginary part of the spectral parameter is not zero, eye-shaped rogue waves appear in one component, and the twisted rogue wave pairs in the other component. As the spectral parameter is real, we derive distinct forms of vector localized and periodic waves on the non-zero background, such as the vector solitons, positons, periodic waves, breathers on the periodic wave background, and rational solitons. These results may be valuable in this investigation of nonlinear waves in physical systems. [ABSTRACT FROM AUTHOR]

This article is about, the new analytical wave solitons, breather-wave, lump-kink and interaction between lump-kink and rogue wave solutions of Laksmanan–Porsezian–Daniel equation with parabolic law along a new definition of derivative has been explored. For this purpose, the exp a function, the extended sinh-Gordon equation expansion scheme and Hirota bilinear schemes have been utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by Mathematica and Maple tools. Our gained solutions are newer than the existing results in the literature. Some of the gained results are explained by 2-dimensional, 3-dimensional and contour plots. Results may be useful for the progress of the model in further study. The schemes used in this research are effective, easy and reliable to handle the other fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process. [ABSTRACT FROM AUTHOR]

This article investigates the optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation with the help of the enhanced modified extended tanh method (eMETEM). The equation is crucial for simulating the bending of light beam, rogue waves in the seas, and fiber pulses in optics. Some different kinds of solitons such as singular-periodic, dark and singular for the KMN equation have been successfully obtained by the used method. Utilizing Maple, we confirm that all acquired solutions satisfy the considered equation. We have also analyzed the impact of the parameters graphically in the considered equation. The results shown in the figures reveal that the parameters of the considered equation have an important effect upon soliton dynamics. It is anticipated that the gained solutions and used approach would create remarkable chances for future work in a variety of science areas. [ABSTRACT FROM AUTHOR]

A (3+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama equation is considered systematically. N-soliton solutions are obtained using Hirota's bilinear method. The employment of the complex conjugate condition of parameters of N-soliton solutions leads to the construction of breather solutions. Then, the lump solution is obtained with the aid of the long-wave limit method. Based on the transformation mechanism of nonlinear waves, a series of nonlinear localized waves can be transformed from breathers, which include the quasi-kink soliton, M-shaped kink soliton, oscillation M-shaped kink soliton, multi-peak kink soliton, and quasi-periodic wave by analyzing the characteristic lines. Furthermore, the molecular state of the transformed two-breather is studied using velocity resonance, which is divided into three aspects, namely the modes of non-, semi-, and full transformation. The analytical method discussed in this paper can be further applied to the investigation of other complex high-dimensional nonlinear integrable systems. [ABSTRACT FROM AUTHOR]

In this study, we build multi-wave solutions of the KdV-5 model through Hirota's bilinear method. Taking complex conjugate values of the free parameters, various colliding exact solutions in the form of rogue wave, symmetric bell soliton and rogue waves form; breather waves, the interaction of a bell and rogue wave, and two colliding rogue wave solutions are constructed. To explore the characteristics of the breather waves, localized in any direction, the higher-order KdV-5 model, which describes the promulgation of weakly nonlinear elongated waves in a narrow channel, and ion-acoustic, and acoustic emission in harmonic crystals symmetrically is analyzed. With the appropriate parameters that affect and manage phase shifts, transmission routes, as well as energies of waves, a mixed solution relating to hyperbolic and sinusoidal expression are derived and illustrated by figures. All the single and multi-soliton appeared symmetric about an axis of the wave propagation. The analyzed outcomes are functional in achieving an understanding of the nonlinear situations in the mentioned fields. [ABSTRACT FROM AUTHOR]

The aim of the manuscript is to study new optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation via the ( G ′ / G , 1 / G )-expansion technique and the exponential rational function (ERF) procedure. The results are produced under the constraint conditions, and their graphical representation highlights them. These discoveries might aid in the comprehension of intricate nonlinear phenomena and oceanography. Studying the investigated equation in this paper is very important in explaining oceanographic phenomena such as rogue waves' ocean currents. [ABSTRACT FROM AUTHOR]

In this paper, we present a modified version of the potential-KdV equation by adding a new stochastic term. The new stochastic potential-KdV describes the propagation of nonlinear optical solitons and photons and appears in the applications of electric-circuits and multi-component plasmas. By using the Cole-Hopf transformation and Hirota bilinear method, we derive novel multi-solitons, lumps, and breather wave-solutions to the proposed model. Also, we provide some graphical analysis to study the impact of the model's coefficients on the propagation of the recovery solutions. Finally, all the reported solutions in this work are checked by direct substitution in the governing equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Sharma-Tasso-Olver-Like equation (STOLE) describing the dynamical behaviour of nonlinear dispersive waves in inhomogeneous medium. By using the Hirota bilinear method, some new type of solitons like breather-wave, kink solitary wave and rogue wave, one-, two- and new three-wave solutions to the STOLE have been determined. These results are achieved and verified by using the Maple software. The obtained results are new from the existing results. For the further explanation of these solutions, different kinds of graphs are also drawn. These solitons suggest that this method is effective, straight forward and reliable as compare to other methods. [ABSTRACT FROM AUTHOR]

In this article, we prove that a sum of solitons and breathers of the modified Kortewegâ€"de Vries equation (mKdV) is orbitally stable. The orbital stability is shown in H 2. More precisely, we will show that if a solution of mKdV is close enough to a sum of solitons and breathers with distinct velocities at t = 0 in the H 2 sense, then it stays close to this sum of solitons and breathers for any time t â©ľ 0 in the H 2 sense, up to space translations for solitons or space and phase translations for breathers, provided the condition that the considered solitons and breathers are sufficiently decoupled from each other and that the velocities of the considered breathers are all positive, except possibly one. The constants that appear in this stability result do not depend on translation parameters. From this, we deduce the orbital stability of any multi-breather of mKdV, provided the condition that the velocities of the considered breathers are all positive, except possibly one (the condition about the decoupling of the considered solitons and breathers between each other is not required in this setting). The constants that appear in this stability result depend on translation parameters of the considered solitons and breathers. [ABSTRACT FROM AUTHOR]

This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model. [ABSTRACT FROM AUTHOR]

[Display omitted] • The multiple Exp-function scheme is employed for searching the multiple soliton solutions for the fractional generalized Calogero-Bogoyavlenskii-Schi-Bogoyavlensky-Konopelchenko (gCBS-BK) equation. • Also, the Hirota bilinear technique is utilized to detecting the lump and interaction with two stripe soliton solutions. • We have obtained more than twelve sets of solutions including a combination of two positive functions as polynomial and two exponential functions. • The semi-inverse variational principle will be used for the considered equation. The graphs for various fractional-order are designed to contain three-dimensional, density, and y-curves plots. • The dynamical structures of these gained lump and its interaction soliton solutions are analyzed and indicated in figures by choosing suitable amounts. The multiple Exp-function scheme is employed for searching the multiple soliton solutions for the fractional generalized Calogero-Bogoyavlenskii-Schiff-Bogoyavlensky- Konopelchenko equation. Moreover, the Hirota bilinear technique is utilized to detecting the lump and interaction with two stripe soliton solutions. The multiple Exp-function scheme and also, the semi-inverse variational principle will be used for the considered equation. We have obtained more than twelve sets of solutions including a combination of two positive functions as polynomial and two exponential functions. The graphs for various fractional-order α are designed to contain three dimensional, density, and y -curves plots. Then, the classes of rogue waves-type solutions to the fractional generalized Calogero-Bogoyavlenskii-Schiff-Bogoyavlensky- Konopelchenko equation within the frame of the bilinear equation, is found. Finally, a direct method which is called the semi-inverse variational principle method was used to obtain solitary waves of this considered model. These results can help us better understand interesting physical phenomena and mechanism. The dynamical structures of these gained lump and its interaction soliton solutions are analyzed and indicated in graphs by choosing suitable amounts. The existence conditions are employed to discuss the available got solutions. [ABSTRACT FROM AUTHOR]

Lump and their interactions with kink, periodic and rogue waves, and periodic cross lump waves will be studied for fifth-order variable coefficient nonlinear-Schrödinger equation in this paper. With the combinations of bilinear, exponent, and trigonometric functions, we'll study different lump soliton solutions. With interaction phenomenon we'll set up some new analytical solutions and also represents them in graphical ways. [ABSTRACT FROM AUTHOR]

The order- n periodic solutions for the (2+1)-D complex modified Korteweg–de Vries (cmKdV) equations are investigated with the aid of Darboux transformation (DT) method. By using Taylor expansion considering the limits λ 2 k − 1 → λ 0 = − a 2 + c i (k = 1 , ... , n) , order-n rational solutions are obtained, among which the order-1 and order-2 solutions are analyzed in detail. By varying different parameter l 1 , two kinds of rational solutions are deduced, namely, the line rogue wave solutions and the lump solutions. Dynamical properties of these solitons, including speed, amplitude, and extreme values, are investigated. It is shown that the line rogue wave solutions appear and disappear, while the lump solutions are localized traveling wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated. • Construction the rational soliton and higher-order RW solutions for the nonlocal LPD equation by the generalized DT. • Dynamic behavior analysis for the singular and nonsingular rational solutions. • The growing compressed effects of the nonsingular RW solutions are analyzed through numerical plots. [ABSTRACT FROM AUTHOR]

For the modified Korteweg–de Vries–Sine Gordon (MKdV-SG) system, we have investigated the degenerate N -soliton solution and the degenerate higher-order breather solution via the bilinear method. The dynamic properties of these degenerate solutions are not only visually displayed through exquisite images, but also accurately described by the specific mathematical expressions. In addition, this paper also intuitively shows how to derive higher-order degenerate solutions from the N -soliton solution for the MKdV-SG equation. [ABSTRACT FROM AUTHOR]

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright-soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis. Breather and rogue wave solutions are constructed under non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution. • Darboux transformation for the coupled complex short pulse equation is derived. • The inverse scattering analysis for the vanishing background is given. • The modulational instability analysis for the plane wave is obtained. • The multi-hump solitons, high order solitons and rogue waves are constructed. [ABSTRACT FROM AUTHOR]

In this work, we derive a family of high order rational solitons of the 3-wave resonant interaction (3WRI) equation by utilizing generalized Darboux-dressing transformation. We observe that breathers in the three components display self-similar behaviors in the process of propagation. For the first order rational soliton, it is interesting that the dark soliton component bifurcates so as to generate a peak whose amplitude is three times higher than the surrounding background itself with an infinite life time. It would provide us a way to generate 'eternal' large amplitude wave in weakly nonlinear dispersive media. Furthermore, the interactions of two rational solitons are studied. • High-order rational solitons of 3WRI are derived by using DT. • Dark soliton bifurcates a peak whose amplitude is three times. • The technique provides a way to generate 'eternal' large amplitude wave. • The interactions of two rational solitons are studied. [ABSTRACT FROM AUTHOR]

We introduce optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation by using the Sardar subequation (SSM) and the new Kudryashov methods (nKM). The KMN equation plays an important role in modeling the fiber pulse in optics, the rogue waves in the oceans and the bending of the light beam. In order to motivate and contribute to researchers working in these fields, we focus on new solutions for the KMN model in fiber pulse transmission by utilizing the Sardar subequation and recently presented New Kudryashov technique. To reach our goal, firstly, the KMN equation has been converted to a nonlinear ordinary differential equation (NODE), then in accordance with the definitions of the proposed methods the solution sets and the solution functions have been obtained. To simulate the new solutions we have obtained and make them more understandable, the topological, anti-topological, periodic,singular and periodic soliton solutions and their properties are presented by 3D and 2D graphics. According to our literature research, these two methods that we are working on have not been applied to the KMN equation before, and we believe that the new solutions we have obtained will be useful to researchers working in modeling in this field. [ABSTRACT FROM AUTHOR]

In this paper, we derive the determinant representation of higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation by using the Darboux transformation and some limit technique. Based on the asymptotic balance between different algebraic terms, we obtain the asymptotic expressions of algebraic soliton solutions with the order 2 ≤ N ≤ 4. It turns out that all the asymptotic solitons have the same amplitudes, most of them are located in the parabolic curves and thus have the varying velocities with the rate O ( | t | − 1 2 ) (except that one pair of asymptotic solitons are located in the straight lines for the odd-order cases), and they exhibit the elastic interactions of the attractive type. In addition, we reveal that the transient rogue waves are generated in the soliton-interaction region and the peak value is exactly N times the amplitude of individual soliton. • The determinant representation of higher-order algebraic soliton solutions is obtained for the Gerdjikov–Ivanov equation. • The asymptotic analysis method is developed on the balance between different algebraic terms up to the subdominant level. • The asymptotic solitons exhibit the attractive interactions and most of them are located in the parabolic curves. • The transient rogue waves can be generated in the soliton-interaction region. [ABSTRACT FROM AUTHOR]

In this paper, a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation is studied. By adding some new constraints to the N -soliton solutions, the resonance Y -type soliton and the interaction solutions between M -resonance Y -type solitons and P -resonance Y -type solitons are obtained. The new types of solutions including the double openings of resonance Y -type soliton solutions and mixed solutions of resonance solitons and breathers are presented. The hybrid solutions consisting of the resonance Y -type solitons and the multiple lump waves are constructed by using restrictions and the long wave limit method. The trajectories of lump waves in hybrid waves are analyzed from the point of view of mathematical mechanism. The results are helpful to explain some new nonlinear phenomena in the theory of solitons. [ABSTRACT FROM AUTHOR]

In this paper, the dimensionless n -component Hirota (alias the n -Hirota) equation is investigated, which describes the wave propagations of n ultrashort optical fields in a fiber. Starting from the modified Darboux transform and its Lax pair with initial non-zero plane-wave conditions, we find the novel multi-parametric families of rational vector rogue wave (RW) solutions for the n -Hirota equation. Furthermore, some weak and strong interactions of rational vector RWs are exhibited for the n -Hirota equation with n = 2 , 3 , 4 , 5 , 6 in detail. In particular, we also deduce the rational vector W -shaped dark and bright solitons of the n -component complex mKdV equation, whose representative wave structures are illustrated for n = 2 , 3 , 4 , 5. Finally, the effect of a small non-integrable deformation of the 3-Hirota equation is explored numerically on the excitation of vector RWs in terms of the Fourier spectral method. These obtained rational vector RW and W-shaped soliton solutions will be useful to further explore the related nonlinear wave phenomena in the sense of multi-component physical systems with non-zero backgrounds. • The novel rational vector RWs are found for the n-Hirota equation. • Some weak and strong interactions of rational vector RWs are exhibited. • Rational vector W -shaped solitons are obtained for the n-cmKdV equation. • Effect of a small deformation of 3-Hirota system is explored on the RW excitations. [ABSTRACT FROM AUTHOR]

In this letter, a large variety of solutions for the fifth-order modified Korteweg–de Vries equation is investigated by the N -soliton solution in bilinear form. The breather and double-pole solutions are derived from the two-soliton solution by different choices for two wavenumbers. Elastic interaction solutions of the breather–soliton and breather–breather are respectively constructed from the three- and four-soliton solutions. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the generalized (2+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. Based on the bilinear Hirota method, the M -lump solution and N -soliton solution are constructed by giving some special activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained with the aid of Maple software. Then, by employing the long wave limit method to the N -soliton solutions, M -lump solutions including 1-lump, 2-lump and 3-lump and the hybrid solutions between lump and solitons and between M -lump and soliton were obtained. Finally, via symbolic computation, their dynamic structures and physical properties were vividly shown by plotting different three-dimensional designs, two-dimensional designs, density designs. These solutions have greatly enriched the exact solutions of (2+1)-dimensional generalized (2+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation on the existing literature. [ABSTRACT FROM AUTHOR]

Using a bilinear transformation and the variable separation approach, The abundant localized structures for the (2+1)-dimensional NLS equation of PT-symmetric media are explored. And Some special soliton solutions such as periodic soliton solution, dromion solutions, ring, peak, breathers and instantons type of solitons are presented by selecting the different functions appropriately. and then, the combined spatial soliton and breather solutions of the (2+1)-dimensional coupled inhomogeneous PT- symmetric nonlinear media with variable coefficients are also obtained, The propagation dynamics of combined solitons are investigated analytically and numerically, and the simulation results show they are applicable for certain potential applications in nonlinear optics and plasma physics. [ABSTRACT FROM AUTHOR]

In this paper, we will focus on the Myrzakulov–Lakshmanan-IV (ML-IV) equation. Based on relevant Lax pair, N -fold generalized Darboux Transformation (DT) will be constructed. Some soliton, breather and rogue wave solutions will be derived under different backgrounds through the odtained DT. In the meantime we will analyze the dynamic features of those solutions graphically. • Construct the N -fold Darboux Transformation. • Obtain a variety of types of exact solutions including soliton solutions, breather solutions and rogue wave solutions. • Analyze dynamic behaviors of those solutions. [ABSTRACT FROM AUTHOR]