Narrow Results

Fluids are common in nature, the study of which helps the design of the related industries. Under investigation in this letter is the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Pfaffian solutions have been obtained based on the Pfaffian technique with the assistance of the real auxiliary function $\phi (y)$. N-soliton solutions with $\phi (y)$ are constructed, where y is the scaled space coordinate and $\phi (y)$ is the steady stream function in the irrotational incompressible flow. Background shapes of the solutions are affected by $\phi (y)$, but the structures of the solutions are affected by the derivative of the log terms in the solutions. Neighborhoods at the origins of the solitons are different in consequence of the different values of the real auxiliary parameter a. One- and two-soliton solutions are illustrated, which are the superpositions of the two kink solitons and different forms of $\phi (y)$. Interactions of the two solitons are presented, from which we see that the velocities, amplitudes and shapes of the two solitons remain unchanged before and after each interaction.

In this paper, we consider the Riemann–Hilbert (RH) method for a fourth-order nonlinear Schrödinger (NLS) equation, which is reduced on the basis of the generalized Davydov’s model by selecting some special parameters. On the basis of the spectral analysis for the Lax pair of the equation, the RH problem is presented. Through a specific RH problem in the sense of irregularity, the multi-soliton solutions are also obtained. In addition, dynamic behaviors of these soliton solutions are given to illustrate the soliton characteristics.

In this study, several novel solutions of strain wave equation for micro-structured solids are obtained by using the powerful modified extended direct algebraic method. In micro-structured solids, the strain wave dynamical equation is utilized to model the wave propagation in micro-structured materials. We have profitably obtained exact traveling wave solutions, including solitons, solitary waves, kink and anti-kink solitary waves, periodic and rational solutions have been found. This approach can not only produce the same solution, but also find new solutions that we think other researchers have missed. These attained solutions help researchers for knowing the physical phenomena of this dynamical equation. The three-dimensional maps of some solutions obtained in this study are plotted through giving appropriate values of the parameters engaged in the solutions for knowing the physical interpretation. The computational work and obtained results demonstrate that the new extended technique, gives an efficient and direct mathematical tools for resolving the nonlinear models in engineering and mathematical physics.

In this paper, we use the new defined direct algebraic method based on some particular Riccati equations to find exact solutions to a Kaup–Newell model equation. Several new solutions which represent long waves parallel to the magnetic fields have been obtained. Many solutions in generalized hyperbolic and triangular function forms, exponential or logarithmic, are expressed explicitly. Most of the solutions determined in the study are new in the related literature.

This paper deals with $M$-soliton solution of the (2+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation by virtue of the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic and cross-kink solutions, which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the $M$-soliton solutions, all cases of the periodic and cross-kink solutions can be captured from the one-, two- and three-soliton solutions. The obtained solutions are extended with numerical simulation to analyze graphically, which results into one-, two- and three-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

Nonlinear Schrodinger equations can model nonlinear waves in plasma physics, optics, fluid and atmospheric theory of profound water waves and so on. In this work, the $exp(-\phi (\xi ))$-expansion, the sine–cosine and Riccati–Bernoulli sub-ODE techniques have been utilized to establish solitons, periodic waves and several types of solutions for the coupled nonlinear Schrödinger equations. These methods with the help of symbolic computations via Mathematica 10 are robust and adequate to solve partial differential nonlinear equations in mathematical physics. Finally, 3D figures for some selected solutions have been depicted.

In this paper, some new traveling wave solutions to the resonant nonlinear Schrödinger’s equation (R-NLSE) with time-dependent coefficients are constructed. The well-known auxiliary equation method is applied to develop numerous interesting classes of nonlinearities, namely the Kerr law and parabolic law. Such approach provides an extensive mathematical tool to develop a family of traveling wave solutions such as bright, dark, singular and optical solutions to the nonlinear evolution model. Moreover, with the aid of symbolic computation the three-dimensional plot and contour plot have been carried out to demonstrate the dynamical behavior of the nonlinear complex model.

The Burgers-type equations are applied to oceanography, hydrodynamic turbulence, gas dynamics, shock-wave formation, acoustic transmission structure, boundary-layer behavior, continuum-traffic simulation, convection-dominated diffusion, wave formation in the thermo-elastic media, vorticity transport, dispersion in the porous media, particle sedimentation in fluid suspension, colloid evolution, and so forth. Hereby, taking into account the wave processes in hydrodynamics and acoustics, we investigate an extended coupled (2+1)-dimensional Burgers system, and with symbolic computation, work out a scaling transformation, two hetero-Bäcklund transformations and two auto-Bäcklund transformations, with the soliton solutions. Our results are dependent on the coefficients in the system.

Recently spin-tensor–momentum coupling (STMC) has been proposed to realize in the Bose–Einstein condensate (BEC). Here, we study the bright soliton in spin-1 three components BEC with STMC. The properties of the bright solitons are discussed by the analytical solutions and numerical solutions. In addition, we also study its dynamic evolution and discover the spontaneous motion, which is different from the spin-orbit-coupled solitons.

Under investigation in this paper are new novel coherent structures of two-dimensional lump-soliton for the Mel’nikov equation. The Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction method are applied to construct a particular family of determinant semi-rational solutions exhibiting various coherent waves to the Mel’nikov equation. We first investigate some novel coherent waves, $N$th-order lumps first appear from the $(N+1)$ dark line solitons and finally disappear into those $(N+1)$ dark line solitons after living on the constant background for a very short period. In contrast to the usual lump, those lumps in the coherent structures of lump-soliton are not only localized in two-dimensional space and but also localized in time.

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

Nonlinear evolution equations of arbitrary order bearing a significantly broad range of capability to illustrate the underlying behavior of naturalistic structures relating to the real world, have become a major source of attraction of scientists and scholars. In quantum mechanics, the nonlinear dynamical system is most reasonably modeled through the Schrödinger-type partial differential equations. In this paper, we discuss the (2+1)-dimensional time-fractional nonlinear Schrödinger equation and the (1+1)-dimensional space–time fractional nonlinear Schrödinger equation for appropriate solutions by means of the recommended enhanced rational $(G^{\prime }/G)$-expansion technique adopting Cole–Hopf transformation and Riccati equation. The considered equations are turned into ordinary differential equations by implementing a composite wave variable replacement alongside the conformable fractional derivative. Then a successful execution of the proposed method has been made, which brought out supplementary innovative outcomes of the considered equations compared with the existing results found so far. The well-generated solutions are presented graphically in 3D views for numerous wave structures. The high performance of the employed technique shows the acceptability which might provide a new guideline for research hereafter.

The Lie symmetry method is used to obtain a variety of closed-form wave solutions for the extended (3+1)-dimensional Jimbo–Miwa (JM) Equation, which describes certain interesting higher-dimensional waves in ocean studies, marine engineering, and other fields. By applying the Lie symmetry technique, we explicitly investigate all the possible vector fields, commutation relations of the considered vectors, and various symmetry reductions of the equation. Based on three stages of Lie symmetry reductions, the JM equation is reduced to several nonlinear ordinary differential equations (NLODEs). Consequently, abundant closed-form wave solutions are achieved, including arbitrary functional parameters. Evolutionary dynamics of some analytic wave solutions are demonstrated through three-dimensional plots based on numerical simulation. Consequently, singular soliton, kink waves, periodic oscillating wave profiles, combined singular soliton profiles, curved-shaped multiple solitons, and periodic multiple solitons with parabolic wave profiles are demonstrated by taking advantage of symbolic computation work. The obtained analytical wave solutions, which include arbitrary independent functions and other constants of the governing equation, could be used to enrich the advanced dynamical behaviors of solitary wave solutions. Furthermore, the study of conservation laws is investigated via the Ibragimov technique for Lie point symmetries.

This paper studies the Hirota–Maxwell–Bloch (H–MB) system and its nonlocal form. Based on the Darboux Transformations (DTs), for H–MB system, we present general double breathers, what is more, we take appropriate modulation frequency and position parameters to investigate the generative mechanism of rogue wave sequences and different periodic breather sequences. For nonlocal Hirota–Maxwell–Bloch (NH–MB) system, we discuss symmetry preserving and broken soliton solutions under zero background. Besides, we present nine combinations of dark and antidark soliton solutions under continuous waves background when PT-symmetry is broken.

This research finds an equation in a continuous domain and a discrete equation governing the system of cold bosonic atoms (CBA) in a zigzag optical lattice using a continuum approximation. Many solutions to the equation were obtained using two distinct methods: the three-wave approach (multi-wave interaction, rational solutions, and rational solution interaction) and the extended sub-equation method. These analytical approaches are more effective, consistent, and comprehensive mathematical tools for obtaining various exact closed-form solutions for a wide range of fractional space-time nonlinear evolution equations encountered in optical physics, condensed matter physics, and plasma physics. The solutions generated are in the form of hyperbolic and trigonometric solutions, and other-form solutions are obtained. Three-dimensional graphics and contour plots are often used to depict the graphical representations of the combined soliton solutions. These findings will aid our understanding of the dynamics of the zigzag optical grids and many other structures formed by colder bosonic atoms. The applied approaches are more simple, efficient, and straightforward to obtain the closed-form solutions for various nonlinear evolution equations in the fields of nonlinear sciences and physical engineering.

This paper studies the ($2+1$)-dimensional modified Heisenberg ferromagnetic (MHF) system, which describes the different nonlinear dynamical structures for long water-waves of small amplitude with weakly nonlinear restoring forces and high-frequency dispersion. This MHF system arises in the propagation of the magnetization vector of the isotropic ferromagnet and biological pattern formation. By employing the Lie symmetry analysis method, infinitesimal generators, Lie point symmetries, potential vector fields, commutation relations of infinitesimal vectors, and attractive symmetry reductions are derived. Based on the two phases of Lie similarity reductions, ($2+1$)-dimensional MHF system is reduced to various nonlinear ordinary differential equations (ODEs). Afterwards, with the help of symbolic computation, we solve the acquired ODEs and obtain a variety of exact closed-form solutions involving arbitrary independent functions and other constant parameters. The physical features of the obtained multi-wave soliton solutions are demonstrated to analyze the impact of the involved arbitrary independent functions on the dynamics of the solitary wave solutions via three-dimensional graphics. These exact solutions are accomplished in the shapes of single solitons, doubly solitons, multi-wave solitons, elastic behavior of multisoliton, oscillating multi-solitons, curved-shaped periodic solitons, and kink-type solitons, and so on. The newly constructed results show the trustworthiness, reliability, and efficiency of the Lie symmetry technique for obtaining the invariant closed-form solutions to nonlinear governing model. Moreover, conservation laws and self-adjoint systems have been obtained by implementing Noether’s technique. By using Lie symmetry analysis, the achieved outcomes might be helpful to understand the physical formation of this model and confirm the effectiveness and authenticity of the mentioned method.

This work attempts to apply the Lie symmetry approach to an updated (2+1)-dimensional KdV equation, recently updated in A.-M. Wazwaz, Nucl. Phys. B 954 (2020) 115009. The equation can be considered as one of the famous examples of the soliton equation. The infinitesimal generators for the governing equation have been found using the invariance property of Lie groups. The commutator table, adjoint table, invariant functions and one-dimensional optimal system of subalgebras are then derived using Lie point symmetries. Some group invariant solutions are derived based on various subalgebras, symmetry reductions and an optimal system. To demonstrate the physical acceptability of the results, the obtained solutions are evaluated using numerical simulation.

The integrable Lakshmanan–Porsezian–Daniel (LPD) equation originating in nonlinear fiber is studied in this work via the Riemann–Hilbert (RH) approach. First, we give the spectral analysis of the Lax pair, from which an RH problem is formulated. Afterwards, by solving the special RH problem with reflectionless under the conditions of irregularity, the formula of general $N$-soliton solutions can be obtained. In addition, the localized structures and dynamic behaviors of the breathers and solitons corresponding to the real part, imaginary part and modulus of the resulting solution $r(x,t)$ are shown graphically and discussed in detail. Unlike 1- or 2-order breathers and solitons, 3-order breathers and soliton solutions rapidly collapse when they interact with each other. This phenomenon results in unbounded amplitudes which imply that higher-order solitons are not a simple nonlinear superposition of basic soliton solutions.

In this paper, modified equal width Burgers’ equation has been investigated with the aid of unified method and bifurcation. This model has many applications in long wave transmission with dispersion and dissipation in nonlinear medium. The applied technique is efficient to retrieve exact solutions and their dynamic behaviors. The obtained solutions are polynomial and rational function solutions. The behavior of dynamical planer system has been analyzed by assigning different values to the parameters, also each possible case has been shown as phase portraits in this research paper. The estimated solutions demonstrate that the proposed approaches are simple, practical, and promising for investigating further equal width equation’s soliton wave solutions and phase portraits.

In this paper, we study the resonant collisions among different types of localized solitary waves in the (2+1)-dimensional Hirota–Satsuma–Ito equation, which are described by N-soliton solutions constructed using bilinear method. Through the asymptotic analysis and limit treatment of the phase shift of these localized waves, the elastic collisions among different localized waves can be transformed into resonant collisions. Hereby, we study the resonant collision between a breather/ lump and a bright line soliton and find two collision situations: (i) the breather is semi-localized in space and the shape of the breather is not localized during the propagation and (ii) the lump wave generates from the bright line wave. At the same time, we investigate the resonant collision between a breather/lump and two bright line solitons. In these evolution processes, we also gain two dynamical behaviors: (iii) the breather is always localized in space and the shape of the breather is not localized during the propagation, and (iv) the lump wave appears from a bright line soliton and then disappears into the other bright line soliton. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in researching the physical phenomena in shallow water waves and nonlinear optics.