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In this investigation, the (3+1)-D potential Yu–Toda–Sasa–Fukuyama (YTSF) equation that arises in physical dynamics is studied for passing the painlevé test and obtaining many various exact solutions. The governing equation has many applications in fluid mechanics. Firstly, we applied the painlevé property for the governing equation and proved that the equation passes the painlevé test. After that, we utilized symmetry analysis to convert the governing equation to various ordinary differential equations. Subsequently, we obtained a new type of exact solutions for YTSF equation by using an Algorithm–Riccati method. The obtained solutions contained several arbitrary constants and functions that enhance the dynamic behaviors of these solutions. The obtained solutions include hyperbolic and trigonometric functions and represent kink wave, singularity wave and solitary wave solutions. [ABSTRACT FROM AUTHOR]

This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering. [ABSTRACT FROM AUTHOR]

The study aims at different families of analytical solutions and their dynamics for the ( 2 + 1 )-dimensional extended Boiti–Leon–Manna–Pempinelli (eBLMP) problem, which is widely used in the fields of physics such as non-linear optics, fluid dynamics, mathematical physics, plasma physics and quantum mechanics. The paper utilises two recently developed efficient mathematical methods: the generalised exponential rational function (GERF) method and the generalised Kudryashov (gK) method. These two methods are versatile, simply applicable to enlighten the new non-linear waveforms. Consequently, these discoveries enhance our understanding of complex systems like ( 2 + 1 )-dimensional eBLMP in the realm of non-linear science. [ABSTRACT FROM AUTHOR]

The interaction between a Riemann wave propagating along the y-axis and a long wave along the x-axis results in a generalised breaking soliton (gBS) equation. Lie symmetries of the equation are generated in this article to derive some rarely available classes of invariant solutions. The presence of arbitrary functions in each solution opens up a broad class of solution profiles. 3D profiles are used to explore more properties of the solutions to the gBS equation. The profiles describe doubly solitons, annihilation of parabolic, periodic solitons, line solitons and solitons on curved surface types. Solution profiles are useful in optical fibre, acoustic waves in a crystal lattice, long waves in stratified oceans, long-distance transmission and shallow water waves. The Lie symmetry approach has future scope to provide more variety in solutions due to the capability of solutions to include functions and arbitrary constants. This research effectively demonstrates the uniqueness of the solutions when compared with the previously published result. Moreover, the adjoint equation and conserved vectors are determined using Noether’s theorem. [ABSTRACT FROM AUTHOR]

We investigated the extended sixth-order Korteweg–de Vries (KdV6) equation which describes many nonlinear phenomena. The improved modified (IM) extended tanh-function method is used to conduct the study. Many exact solutions, like dark, combo singular-dark and singular soliton are achieved. Moreover, hyperbolic solutions, singular periodic solutions, rational solutions, exponential solutions and Jacobi elliptic function (JEF) solutions are derived. Graphical illustrations are presented to visually depict the dynamics of selected solutions. [ABSTRACT FROM AUTHOR]

In this work, the exact travelling wave solutions to the forced Korteweg–de Vries (fKdV) equation with different force terms are studied with the help of symbolic computations. This equation is derived from a straightforward mathematical model that describes the behaviour of a shallow fluid layer when influenced by external forces. The fKdV equation has many applications in diverse fields, including fluid dynamics, plasma physics, soliton theory and mathematical physics, for modeling wave propagation and nonlinear phenomena under the influence of external forces. Solitary wave solutions for this equation have been derived using three distinct techniques: the extended (G ′ / G) -expansion method, the Kudryashov method and the (1 / G ′) -expansion method. As a result, several new solutions have been achieved which are in the form of hyperbolic, trigonometric, rational and exponential functions. Finally, the effects of different time-dependent external forces have been studied by presenting 3D, 2D and contour plots. It can be seen that the external forces affect the background and speed of solitary waves. The results could be expected to be helpful in understanding the propagation of solitary waves subjected to external forces. [ABSTRACT FROM AUTHOR]

A new generalised non-local nonlinear Schrödinger (NLS) equation is introduced which possesses a Lax pair and is parity–time (PT)-symmetric. Thus, it is confirmed that the generalised non-local NLS equation is integrable. The inverse scattering transform for the generalised non-local NLS equation is developed using a Riemann–Hilbert problem for rapidly decaying initial data and an approach for finding pure soliton solutions is described. The analytical characteristics of the eigenfunctions, scattering data and their symmetries are discussed. Finally, using Mathematica some important two-dimensional plots of the wave solutions are shown to illustrate the dynamics of the model. [ABSTRACT FROM AUTHOR]

Variable-coefficient equations can be used to describe certain phenomena when the inhomogeneous media and nonuniform boundaries are taken into consideration. It is meaningful to solve the exact solution of variable-coefficient equations. In this paper, a (2+1)-dimensional variable-coefficient Boussinesq equation is investigated. The integrability is firstly examined by the Painlevé analysis method. Secondly, the Bäcklund transformations, one- and two-soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation are studied by virtue of the Hirota bilinear method. Propagation characteristics and interaction behaviors of the solitons are discussed: (i) soliton shapes and interaction behaviors are affected by the variable coefficients, and (ii) the two-soliton interaction is elastic, and the shape and velocity does not change after the collision, and only the shift changes. [ABSTRACT FROM AUTHOR]

The current work is concerned with some novel solutions of the new extended (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation (BLMPE) that plays a key role in the incompressible fluid. The N-soliton solutions (NSSs) are extracted by virtue of the Hirota form which is developed using the Cole–Hopf transform. Based on the NSSs, we elaborate the resonance conditions of the soliton molecules and derive the soliton molecules of the 2-solitons, 3-solitons and 4-solitons on the (x, y)-, (x, z)- and (y, z)-planes. Additionally, we also explore some novel hybrid interaction solutions including the interaction between the soliton molecule and solitons and the interaction between the breather solution and the solitons. Correspondingly, the dynamic properties of the solutions are depicted graphically. The derived solutions in this study are all new and can enlarge the exact solutions of the new extended (3 + 1)-dimensional BLMPE. Besides, they can enable us to understand nonlinear dynamic behaviours better. [ABSTRACT FROM AUTHOR]

The current research aims to look into the exact solutions (ESs) of the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation (BLMPE), which describes the irrotational-incompressible fluid. The Hirota bilinear equation of the problem under consideration is developed by applying the Cole–Hopf transform. Then, the ansatz function, combined with the symbolic computation, is utilised to extract the ESs. Some new types of ESs like the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and the interaction wave solution (IWS) are successfully constructed. Furthermore, ESs such as the travelling wave solution (TWS) are also explored via the modified Kudryashov method. The dynamic characteristics of the ESs are described graphically using the Mathematica software and their physical explanations are presented. Compared with the publicly available literature, the obtained ESs are all new and have not yet been reported. The ideas in this work can be used to investigate the ESs of the other PDEs that appear in physics. [ABSTRACT FROM AUTHOR]

In this study, we take into account the (2 + 1)-dimensional Boussinesq equation, a nonlinear evolution partial differential equation that describes how gravity waves move across the surface of the ocean. The symmetry reductions and group invariant precise solutions are systematically determined using the Lie symmetry analysis. We derive the precise multiple wave solutions using the multiple exp-function method, and then, using the multiplier method, we give the conservation laws. The dynamics of complicated waves and their interplay are faithfully recreated by the findings. [ABSTRACT FROM AUTHOR]

This work aims to explore the optical solitons of the Kaup–Newell model (KNM) which describes the glamorous technology of sub-picosecond pulses that spread via single-mode optical fibres using two innovative methods, namely Wang’s Bäcklund transformation-based method (WBTBM) and Wang’s direct mapping method (WDMM). Different kinds of exact solutions expressed in the form of the rational function, Exp function, sin–cos function, sinh–cosh function, hyperbolic function (sech function and csch function) and trigonometric function (sec function and csc function) are obtained, which include the bright soliton, dark soliton, singular soliton and other travelling wave solutions. To reveal the dynamic behaviours of the different solutions, the 3D plots, 2D contours and 2D curves are drawn with the help of the Mathematica software. The findings in this work confirm that the mentioned methods are effective, straightforward and powerful, and can be used to handle other partial differential equations (PDEs) arising in the optics. [ABSTRACT FROM AUTHOR]

In this article, we focuss primarily on a ( 3 + 1 )-dimensional nonlinear equation in fluids and the dynamic behaviour of the lump solution and new interaction solutions of the equation. First, the bilinear form of the equation is investigated using the Bell polynomials and the Hirota bilinear method. The Bell-polynomial-type Bäcklund transformation of the equation and the corresponding Lax pair are obtained by using the transformation technique. Based on the obtained bilinear form, we use the positive quadratic function method to obtain the lump solutions of the equation and study its dynamic behaviour in detail. On the other hand, we constructed two types of interaction solutions, lump-soliton and lump-periodic, and combined the images to vividly show the interaction phenomenon. Furthermore, the homoclinic breather solutions of the equation are constructed using the homoclinic test method. Finally, we hope that the solutions we obtained can explain some nonlinear phenomena in the fluid mechanics and shallow water wave fields. [ABSTRACT FROM AUTHOR]

In this paper, a reciprocal transformation is applied to connect a negative flow of the Degasperis–Procesi (DP) equation with the Kaup–Kupershmidt (KK) equation. Via Darboux transformation, we obtained multisoliton solutions of a negative flow in the DP hierarchy and its short-wave limit. [ABSTRACT FROM AUTHOR]

The (3 + 1) -dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients, which have very crucial applications in various areas, such as fluid dynamics, plasma physics and so on, are studied via unified and generalised unified method in this work. Solitary, soliton, elliptic, singular (periodic type) and non-singular (soliton-type) solutions of these two equations are extracted using the unified method. Otherwise, polynomial solutions in double-wave form and rational solutions in double-soliton form of the modified KdV equation with variable coefficients are acquired by exploiting the generalised unified method. The dynamical demeanours of these solutions help to comprehend the physical phenomena reflected by the equations, are depicted and analysed graphically for specific values of randomly undetermined parameters, which are diverse in each solution. The outcomes show that these two methods are quite trustworthy and effective to explore numerous solutions of nonlinear partial differential equations. We recognise that two approaches have never been utilised to study these two equations and work carried out in this paper is fresh and handy. Compared to previous methods, more comprehensive solutions can be obtained using them in this paper. [ABSTRACT FROM AUTHOR]

This paper focusses on a class of non-isospectral (2 + 1) -dimensional Calogero–Degasperis system, which is a more general form of the classical nonlinear Schrödinger equation. Primarily, according to the plane wave seed solutions, we analyse the modulational instability of this system and obtain the formation mechanism of different localised waves. Secondly, based on the known Lax pair, we construct the generalised (n , N - n) -fold Darboux transformation of this system. As an application of the resulting Darboux transformation, we not only show the interaction structures of the multisoliton solutions, but also analyse its long-time asymptotic behaviours and list the relevant physical properties. In order to explore the relationship with differential geometry, we also show the multisoliton surfaces. Subsequently, we give some higher-order lump solutions and analyse their large-parameter asymptotic states. We also give some mixed interactional solutions to better understand the interaction phenomena of different localised waves, whose propagation structures and characteristics are shown graphically. These results and phenomena may be helpful to understand some physical phenomena in nonlinear optics, fluids and Bose–Einstein condensates. [ABSTRACT FROM AUTHOR]

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]

This paper is devoted to the study of higher-order soliton solutions and some hybrid behaviours between them in the ( 2 + 1 )-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation. Firstly, we obtain a new lump solution by studying the degenerate behaviour of the breathers. Secondly, we focus on studying the N-soliton solutions, higher-order breathers and higher-order lump solutions by using the long-wave limit method. Some novel higher-hybrid behaviours are studied to show the effect of the superposition of different solitons on the structure of the solution, such as the hybrid behaviours between the m-lump solution and n-soliton, the hybrid behaviours between the 1-breather solution and n-soliton. Finally, many three-dimensional images and density projections of spatial structures are displayed to reflect the evolutionary behaviour of the higher-order soliton solutions with the increase of order number. [ABSTRACT FROM AUTHOR]

This investigation proposes the novelty of the modified (G ′ G 2) -expansion method to look for new exact traveling wave solutions to two important nonlinear evolution equations such as the Konno–Oono equation and the Boussinesq equation. The simplicity and dependability of this approach make it advantageous for solving nonlinear issues. The technique involves wave transformation to get the nonlinear evolution equation down to the corresponding ordinary differential equations. The solutions include some new exact traveling solutions and are categorized into three classes of trigonometric, hyperbolic, and rational solutions. Numerical simulation is used to support the solutions and give them physical meaning. These results contain a large number of travelling wave solutions that are crucial for explaining certain scientific phenomena in fluid media. [ABSTRACT FROM AUTHOR]

This research studies the Kudryashov's equation by using the generalised exponential rational function (GERF) technique. This technique allows us to use the traveling wave reduction for Kudryashov's equation which provides the system of equations associated with the real and imaginary parts of the profile pulse for a complex function. First and foremost, we reduced these systems of equations into nonlinear ordinary differential equations (ODE) under the travelling wave transformations. The critical steps of the GERF technique are then applied to the resulting nonlinear ODE. With the assistance of the mentioned technique, we obtained different kinds of families of optical soliton solutions for Kudryashov's equation. The real part, imaginary part and modulus of some of the obtained solutions are investigated/explored by the three-dimensional visual representations under the appropriate choice of the involved arbitrary parameters with suitable range space, demonstrating that the solutions represent a series of solitons, bright solitons, dark solitons, singular solitons, the interaction of solitons with solitary wave profiles, the interaction of kink waves with solitons and the interaction of kink waves with solitons. [ABSTRACT FROM AUTHOR]

The one-dimensional hyperbolic mean curvature flow for Lagrangian graphs is discussed in this paper. In the beginning, infinitesimal generators, symmetry groups and an optimal system of symmetries for the proposed hyperbolic Lagrangian mean curvature flow are obtained based on the Lie symmetry approach. Additionally, several invariant solutions are discovered using reduced equations. More specifically, we use the power series method to attain explicit solutions. [ABSTRACT FROM AUTHOR]

The paper aims to construct optical solitons and travelling wave solutions to two birefringent nonlinear models which consist of two-component form of vector solitons in optical fibre: the Biswas–Arshed model with Kerr-type nonlinearity and without four-wave mixing terms and the nonlinear Schrödinger equation with quadratic-cubic law of refractive index along with four-wave mixing terms. These nonlinear Schrödinger equations are applied in many physical and engineering fields. Optical solitons are considered in the context of photonic crystal fibres, couplers, polarisation-preserving fibres, metamaterials, birefringent fibres, and so on. Two reliable integration architectures, namely, the extended simplest equation method and the generalised sub-ODE approach, are adopted. As a result, bright soliton, kink and dark soliton, singular soliton, hyperbolic wave, a periodic wave, elliptic function solutions of Weierstrass and Jacobian types, and other travelling wave solutions, such as breather solutions and optical rogons, are derived, together with the existence conditions. In addition, the amplitude and intensity diagrams are portrayed by taking appropriate values for a few selected solutions. Furthermore, based on linear stability analysis, the modulation instability was explored for the obtained steady-state solutions. The reported results of this paper can enrich the dynamical behaviours of the two considered nonlinear models and can be useful in many scientific fields, such as mathematical physics, mathematical biology, telecommunications, engineering and optical fibres. This study confirms that the proposed approaches are sufficiently effective in extracting a variety of analytical solutions to other nonlinear models in both engineering and science. [ABSTRACT FROM AUTHOR]

This article investigates a sixth order integrable nonlinear partial differential equation model that fulfills the Hirota N-soliton. Space and time-dependent shift, rotation and space-dependent shift, time and space translations, and time and space dilations Lie point symmetries are presented methodically. Under a specific point symmetries, the Lie point symmetries lead to group invariant solutions. The significance of conservation laws of the underlying equation are shown. The results are quite accurate in recreating complex waves and the dynamics of their interactions. [ABSTRACT FROM AUTHOR]

This article investigates the accuracy of the obtained analytical solutions of the nonlinear fractional Ostrovsky (NLFO.) equation. The researched solutions are obtained by exploiting the well-known generalized Tanh–function (GTF.) method with the Atengana– conformable fractional (ACF.) derivative in the wave transformation. Five–numerical schemes (Adomian decomposition (AD), El Kalla (EK), cubic B-Spline (CBS), extended cubic B-Spline (ECBS), and exponential cubic B-Spline (ExCBS)) handle these solutions to check their solutions. The investigated fractional model is a general form of the KdV equation; however, the KdV equation’s solutions replace this effect with radiating inertia gravity waves. This model also describes a weak description of nonlinear ocean wave processes considering Earth rotation. The obtained analytical and numerical results are sketched through Mathematica 12 in different plot types to explain the waves’ dynamical behavior. Additionally, the computational obtained solutions’ stability is investigated. Finally, the paper’s contribution and obtained results’ novelty are demonstrated by comparing our solutions with those that have been obtained in previous scientific articles.

This study is about some new optical wave solutions to the time-fractional Kudryashov equation. The obtained solutions may be applied to demonstrate the nonlinear fractional model in a better way. These solutions are in the form of dark, singular, singular-dark solitons and other solutions with certain conditions. The obtained results were verified using Mathematica software. Modified integration method, the extended Jacobi’s elliptic expansion functions method, is utilised to secure the aforesaid solutions. These optical solitons suggest that this method is effective, straightforward and reliable compared to other methods. [ABSTRACT FROM AUTHOR]

Belov–Chaltikian (BC) lattice equation, which is related to the research of lattice analogues of W-algebras, is under consideration in this work. This equation may be viewed as an extension of the Volterra lattice equation. Firstly, we correspond BC lattice equation to several continuous equations under the continuous limit. Secondly, based on the known 3 × 3 matrix form Lax pair of this discrete equation, we construct its discrete generalised (m , 3 N - m) -fold Darboux transformation for the first time and successfully popularise this technique from 2 × 2 Lax pair to 3 × 3 Lax pair. Finally, by applying the resulting Darboux transformation, we get its higher-order rational solutions and analyse their singular trajectories and dynamics using the graphics and limit-state analysis. [ABSTRACT FROM AUTHOR]

Under vanishing and non-vanishing boundary conditions, we consider general soliton solutions of a fully PT-symmetric multidimensional non-local nolinear Schrödinger equation with time reversal. Concrete expressions could be written as N × N Gram-type determinants by employing Hirota's bilinearity and the KP hierarchy reduction, for positive integer N. Furthermore, the typical dynamics and behaviours of high-order soliton solutions are illustrated graphically. [ABSTRACT FROM AUTHOR]

This paper studies the optical soliton solutions of the Biswas–Arshed equation with the help of two different techniques, such as the generalised exponential rational function (GERF) technique and Kudryashov's simplest equation technique. The GERF technique extracts distinct families of exact solitary wave solutions involving trigonometric function solutions, hyperbolic function solutions, rational function solutions, etc. After that, we apply Kudryashov's simplest equation method in the context of Bernoulli and Riccati equations to attain different kinds of families of exact soliton solutions. All the acquired solutions of the equation have numerous applications in many branches of nonlinear sciences such as plasma physics, superconductivity, nonlinear optics, biophysics, star formation, quantum mechanics, etc. and many more connected fields of nonlinear wave sciences. The exact solitary wave solutions obtained by GERF technique and Kudryashov's simplest equation technique are in more generalised form as they contained several arbitrary parameters. Subsequently, to understand the behaviour of deduced solutions, we graphically discuss the real part, imaginary part and modulus of these solutions by suitable choice of involved arbitrary parameters. [ABSTRACT FROM AUTHOR]

The Korteweg–de Vries (KdV) equation can describe a weakly nonlinear long wave when its phase speed arrives a wave maximum with the infinite length in shallow water. In this paper, the investigation of a two-dimensional (2D) KdV equation is conducted using the double-function method. A unified solution for separating the variables is given based on the projective Riccati equation in the double-function method and its unified exponential-form solution, which covers many familiar solutions including sinh , cosh , sech , tanh , csch , coth , sec , tan , csc , cot solutions in previous literatures. From this exponential-form solution for the separation of the variables, the imperfect elastic collision between bell-shaped and peak-shaped double-loop semi-folded solitary waves, imperfect elastic collision between peak-shaped double-loop semi-folded and double-loop full-folded solitary waves and perfectly inelastic collision between bell-shaped and peak-shaped double-loop semi-folded solitary waves are graphically and analytically discussed. Moreover, the collision properties between solitary waves are quantificationally analysed by the asymptotic analysis. Phase shifts and their difference values of collisions between double-loop multivalue solitary waves are analytically presented. [ABSTRACT FROM AUTHOR]

The non-local symmetries of the coupled Lakshmanan–Porsezian–Daniel (LPD) equations are obtained with the help of the known Lax pair. By introducing an auxiliary variable, the coupled LPD equations are extended to a closed prolonged system and the non-local symmetries are localised to the Lie point symmetries of the prolonged system. Furthermore, based on the Lie point symmetries of the prolonged system, the exact solutions and non-local conservation laws of the coupled LPD equations are derived. [ABSTRACT FROM AUTHOR]

The current study suggests a new generalisation of highly dispersive nonlinear Schrödinger-type equation (NLSE) with perturbation terms. With polynomial refractive index, known by cubic–quintic–septic (CQS) law and Hamiltonian-type cubic perturbation terms, the new model includes eighth-order dispersion term. The generalised Riccati simplest equation method (RSEM) and the modified simplest equation method (MSEM) are successfully utilised to analytically process the fractional version of the considered NLSE. A diverse collection of bright, dark and singular optical solitons under some constraints, in hyperbolic, periodic and rational-exponential forms are derived. Graphical interpretations of some obtained solutions are displayed. The two considered schemes, with different algorithms, show an influential mathematical tool for processing nonlinear fractional evolution equations. [ABSTRACT FROM AUTHOR]

In the current work, we study the coupled Schrödinger–KdV (CS–KdV) equation. The Lie point symmetry generators are derived for the CS–KdV equation. We also show that the CS–KdV equation is non-linearly self-adjoint and we use this property to construct conservation laws corresponding to the symmetries of the model. Symmetry reduction technique in conjunction with the extended tanh method is used to calculate the solitary wave solution. Also, conservation laws for the CS–KdV equation is derived by applying the new conservation theorem of Ibragimov. The undetermined coefficients and Bernoulli sub-ODE methods are suggested for constructing the topological, non-topological and singular soliton solutions to the model. New breather solitary solutions are constructed for CS–KdV equation by using bilinear form and the extended homoclinic test approach (EHTA). [ABSTRACT FROM AUTHOR]

In this paper, we derive the solitons solutions to the cubic–quartic nonlinear Schrödinger equation (CQ-NLSE) in birefringent fibers with three laws of nonlinearity. These laws are parabolic law, quadratic-cubic law and non-local law. The new extended generalized Kudryashov method and our new method proposed for the first time have been applied. Many solutions have been found. Dark, bright and singular soliton solutions existed under constraint conditions. The comparative analysis of the two proposed methods shows that in the two cases of parabolic and non-local laws, the first method gives the dark and singular solitons in terms of tanh–coth hyperbolic functions, respectively, while the second method gives the bright and singular solitons in terms of sech–csch hyperbolic functions, respectively. But in the case of quadratic-cubic law, both methods give the bright-singular solitons in terms of sech 2 –csch 2 hyperbolic functions, respectively, with different arguments. [ABSTRACT FROM AUTHOR]

The non-local symmetries for the generalised Broer–Kaup (BK) system are obtained with the truncated Painlevé expansion method. The non-local symmetries can be localised to the Lie point symmetries for the extended system by introducing auxiliary dependent variables. The finite symmetry transformations related to the non-local symmetries are computed. The consistent Riccati expansion and consistent tanh expansion integrability of the generalised BK system are proved, and the single soliton, two-resonant soliton and soliton–cnoidal wave interaction solutions are derived. [ABSTRACT FROM AUTHOR]

In this work, our goal is to find more general exact travelling wave solutions of the (1 + 1)- and (2 + 1)-dimensional nonlinear chiral Schrödinger equation with conformable derivative by using a newly developed analytical method. The governing model has a very important role in quantum mechanics, especially in the field of quantum Hall effect where chiral excitations are present. In two-dimensional electron systems, subjected to strong magnetic fields and low temperatures, the quantum Hall effect can be observed. By using the method, called the rational sine-Gordon expansion method which is a generalised form of the sine-Gordon expansion method, we found complex dark and bright solitary wave solutions. These solutions have important applications in the quantum Hall effect. [ABSTRACT FROM AUTHOR]

The aim of this paper was to study a generalised (2 + 1)-dimensional nonlinear Schrödinger equation, which can give us many important mathematical and physical models to illustrate various nonlinear phenomena in physical and engineering sciences. Based on a general Hirota bilinear form, the breather-type and non-singular rational solutions are presented. Akhmediev breather and Ma breather solutions can be considered as solutions representing the nonlinear propagation of the unstable model. In addition, by means of the bilinear transformation method, the fundamental rogue waves are given in terms of the Grammian determinant, which are shown to be line rouge waves. It is then demonstrated that the non-singular rational solutions generated via the long wave limit approach cover the rogue waves presented by the bilinear transformation method. The results presented in this paper exhibit the complexity and diversity of dynamical behaviour for the considered complex system. [ABSTRACT FROM AUTHOR]

The nonlinear crystal with sharp-step dependence of dielectric function on the electric field covered by shielding thin film is considered. A change in the signs of the nonlinearity coefficient is induced by the electric field. Different sets of boundary conditions are used to calculate the exact solutions of the formulated equation. New types of nonlinear surface waves are found and the conditions of their existence are derived analytically. In certain cases, some types of surface waves are characterised by free-varying wave frequency and effective refractive index. Other types of surface waves can propagate with a certain effective refractive index and arbitrary frequencies. In addition, new types of waves propagating only with fixed effective refractive index and fixed frequency are described. [ABSTRACT FROM AUTHOR]

The Lie symmetry technique is utilised to obtain three stages of similarity reductions, exact invariant solutions and dynamical wave structures of multiple solitons of a ( 3 + 1 )-dimensional generalised BKP–Boussinesq (gBKP-B) equation. We obtain infinitesimal vectors of the gBKP-B equation and each of these infinitesimals depends on five independent arbitrary functions and two parameters that provide us with a set of Lie algebras. Thenceforth, the commutative and adjoint tables between the examined vector fields and one-dimensional optimal system of symmetry subalgebras are constructed to the original equation. Based on each of the symmetry subalgebras, the Lie symmetry technique reduces the gBKP-B equation into various nonlinear ordinary differential equations through similarity reductions. Therefore, we attain closed-form invariant solutions of the governing equation by utilising the invariance criteria of the Lie group of transformation method. The established solutions are relatively new and more generalised in terms of functional parameter solutions compared to the previous results in the literature. All these exact explicit solutions are obtained in the form of different complex wave structures like multiwave solitons, curved-shaped periodic solitons, strip solitons, wave–wave interactions, elastic interactions between oscillating multisolitons and nonlinear waves, lump waves and kinky waves. The physical interpretation of computational wave solutions is exhibited both analytically and graphically through their three-dimensional postures by selecting relevant values of arbitrary functional parameters and constant parameters. [ABSTRACT FROM AUTHOR]

Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

This paper deals with the nonlinear generalised advection–diffusion–reaction (GADR) equation subject to some initial and boundary conditions (BCs). Some exact finite difference (EFD) schemes and non-standard finite difference (NSFD) schemes are derived. Positivity and boundedness of the proposed NSFD schemes are analysed analytically and numerically. Theoretical results are very well supported by solved examples. The major achievements of NSFD schemes are that they give highly accurate solutions for very few spatial and temporal divisions. [ABSTRACT FROM AUTHOR]

The dynamics of localised solitary wave solutions play an essential role in the fields of mathematical sciences such as optical physics, plasma physics, nonlinear dynamics and many others. The prime objective of this study is to obtain localised solitary wave solutions and exact closed-form solutions of the (2 + 1) -dimensional universal hierarchy equation (UHE) using the Lie symmetry approach. Besides, the Lie infinitesimals, all the vector fields, commutation relations of Lie algebra and symmetry reductions are derived via the Lie transformation method. Meanwhile, the universal hierarchy equation is reduced into nonlinear ODEs through two stages of symmetry reductions. The closed-form invariant solutions are attained under some parametric conditions imposed on infinitesimal generators. Because of the presence of arbitrary independent functional parameters and other constants, these group-invariant solutions are explicitly displayed in terms of arbitrary functions that are more relevant, beneficial and useful for explaining nonlinear complex physical phenomena. Furthermore, the dynamical structures of the obtained exact solutions are illustrated for suitable values of arbitrary constants through 3D-plots based on numerical simulation. Some of these localised solitary waves are double solitons, periodic lump solitons, dark solitons, five-solitons, hemispherical solitons and lump-type solitons. [ABSTRACT FROM AUTHOR]

The prime objective of this paper is to obtain some new families of exact solitary wave solutions of the Klein–Gordon–Zakharov (KGZ) equations via computerised symbolic computation on Wolfram Mathematica. By applying the generalised exponential rational function method, numerous exact soliton solutions are constructed for the KGZ equations, which provide a model of the interaction between the Langmuir wave and the ion-acoustic wave in high-frequency plasma. Consequently, the exact solitary wave solutions are obtained in different forms of dynamical wave structures of solitons including multisolitons, lump-type solitons, travelling waves, kink waves, also trigonometric and hyperbolic function solutions, and rational function solutions. Moreover, the dynamical behaviour of the resulting multiple soliton solutions is discussed both analytically and graphically by using suitable values of free parameters through numerical simulation. The reported results have rich physical structures that are helpful to explain the nonlinear wave phenomena in plasma physics and soliton theory. [ABSTRACT FROM AUTHOR]

In this article, the Paul–Painleve approach (PPA) which was formulated recently and built on the balance role has been used perfectly to achieve new impressive solitary wave solutions to the nanosoliton of ionic waves (NSOIW) propagating along the microtubules in the living cells. In addition, variational iteration method (VIM) has been applied in the same vein and parallel to establish numerical solutions of this model. [ABSTRACT FROM AUTHOR]

In this paper, multiple rogue wave solutions of a generalised Hietarinta-type fourth-order equation in ( 2 + 1 )-dimensional dispersive waves were studied by applying the bilinear method. We obtained its 1-rogue wave, 3-rogue wave and 6-rogue wave solutions. Similarly, their corresponding maps which can finely explain their physical structure and properties were graphically shown through symbolic computation approach. It is obvious that the centre of the 3-rogue wave possesses a triangular structure while 6-rogue wave has a hexagon structure and they are made of three and six independent 1-rogue waves, respectively. Furthermore, the results obtained have immensely augmented the exact solutions of the generalised Hietarinta-type equation on the available literature and enabled us to understand the nonlinear dynamic system deeply. [ABSTRACT FROM AUTHOR]

Two-layer fluids are seen in fluid mechanics, thermodynamics and medical sciences. Lattices are seen in solid-state physics. In a two-layer liquid or a lattice, a ( 3 + 1 )-dimensional generalised Yu–Toda–Sasa–Fukuyama equation is hereby studied with symbolic computation. Via the Hirota method, bilinear form and bilinear auto-Bäcklund transformation under certain coefficient constraints are obtained. Breather solutions are worked out based on the Hirota method and extended homoclinic test approach. Considering that the periods of breather solutions tend to infinity, we derive the lump solutions under a limit procedure. We observe that the amplitudes of the breather and lump remain unchanged during the propagation. Furthermore, we graphically present the breathers and lumps under the influence of different coefficients in the equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two extended higher-order KdV models (i.e., the extended Sawada–Kotera equation and the extended Lax equation), which can successfully describe propagation of dimly nonlinear long waves in fluids and ion-acoustic waves in harmonic sparklers. First, we present a general formula of multisoliton solutions of the two models. We then build the interaction solutions in terms of hyperbolic and sinusoidal functions by using multisoliton solutions with appropriate complex conjugate parameters controlling the phase shifts, propagation direction and energies of the waves. In particular, we present their collision solutions in the identical plane with different parametric constraints, which degenerate to the line rogue waves, x-shaped rogue waves, cnoidal periodic waves, interactions of rogue and bell waves, line breather and double breather waves. The dynamical characteristics of the wave solutions are shown graphically by choosing some special parameter values. [ABSTRACT FROM AUTHOR]

In this article, we take into account the (1 + 1) -dimensional dispersive long-wave equation which is reduced from the Wu–Zhang equation. Firstly, the B a ¨ cklund transformation and the non-local symmetry are successfully acquired from the truncated Painlev e ´ expansion. At the same time, the non-local symmetry is transformed to Lie point symmetry by a suitable prolonged system. Then some solitary solutions are derived without a hitch via new B a ¨ cklund transformation which originates from Lie point symmetry. Secondly, by utilising the consistent Riccati expansion method and the consistent tanh-expansion method, we find the interaction solutions between soliton and cnoidal wave by using the Jacobi elliptic function. Lastly, the conservation laws which are related to symmetries of the equation are successfully obtained by Ibragimov’s method. [ABSTRACT FROM AUTHOR]

By using the reductive perturbation technique, the nonlinear dust ion-acoustic solitary wave models of the damped Korteweg–de Vries (D-KdV) and modified damped Korteweg–de Vries (D-mKdV) equations are formulated. We constructed the more general and new solitary wave solutions of nonlinear damped KdV and damped mKdV equations by using the modified mathematical technique. These obtained solutions are more useful in the development of quantum plasma, dynamics of solitons, dynamics of adiabatic parameters, dynamics of fluid and problems of biomedical, industrial phenomena. The new solutions are obtained in the shape of dark solitons, bright solitons, traveling wave and kink and anti-kink wave solitons. We show the physical structure of new solutions by two and three-dimensions graphical to know the physical interpretation of different structure of dust ion-acoustic solitary wave. The calculations show that this new technique is more powerful, effective, straightforward, and fruitfulness to study analytically other nonlinear complex physical models in plasma physics, mathematical physics, fluid mechanics, hydrodynamics, mathematical biology and many other physical sciences. [ABSTRACT FROM AUTHOR]