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In this research article, the perturbed nonlinear Schrodinger equation (P-NLSE) is examined by utilizing two analytical methods, namely the extended modified auxiliary equation mapping and the generalized Riccati equation mapping methods. Consequently, we establish several sorts of new families of complex soliton wave solutions such as hyperbolic functions, trigonometric functions, dark and bright solitons, periodic solitons, singular solitons, and kink-type solitons wave solutions of the P-NLSE. Using the mentioned methods, the results are displayed in 3D and 2D contours for specific values of the open parameters. The obtained findings demonstrate that the implemented techniques are capable of identifying the exact solutions of the other complex nonlinear evolution equations (C-NLEEs) that arise in a range of applied disciplines.

The current study deals with exact soliton solutions for Schrodinger-Hirota (SH) equation via two modified integration methods. Those methods are known as the improved ( G ′ / G ) -expansion method and the Kudryashov method. This model is a generalized version of the nonlinear Schrodinger (NLS) equation with higher order dispersion and cubic nonlinearity. It can be considered as a more accurate approximation than the NLS equation in explaining wave propagation in the ocean and optical fibers. A novel derivative operator named as the conformable truncated M-fractional is used to study the above mentioned model. The obtained results can be used in describing the Schrodinger-Hirota equation in some better way. Moreover the obtained results are verified through symbolic computational software. Also, the obtained results show that the suggested approaches have broaden capacity to secure some new soliton type solutions for the fractional differential equations in an effective way. In the end, the results are also explained through their graphical representations.

In this work, the improved modified extended tanh scheme is implemented to extract exact travelling wave solutions for perturbed nonlinear Schrodinger’s equation with Kudryashov’s law of refractive index and dual form of generalized nonlocal nonlinearity. Various types of solutions are extracted such as bright solitons, singular solitons, dark solitons, singular periodic wave solutions, periodic wave solutions, Jacobi elliptic functions, plane wave and hyperbolic wave solutions. Moreover, 3D and contour plots of some solutions are illustrated to show the physical nature of obtained solutions.

Introduction The Gross-Pitaevskii equation is a class of the nonlinear Schrodinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. Objectives We give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. Methods By constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. Results The results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. The results enrich the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

Coupled derivative nonlinear Schrodinger (cDNLS) equations play an important role in plasma physics, optics and other fields. One-order analytical solutions of the cDNLS equations are obtained by virtue of the Darboux transformation. Via adjustment about the parameters, bright-dark conversion mechanism of rogue wave is obtained; combinations of rogue wave and breather/bellshape soliton with different waveform are gotten; distance between rogue wave and breather/bellshape soliton can be changed. In addition, we find that rogue wave has affect on the propagation direction of breather/bellshape soliton.

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

We study the asymptotic linear stability of a two-parameter family of solitary waves for the isothermal Euler–Poisson system. When the linearized equations about the solitary waves are considered, the associated eigenvalue problem in $$L^2$$ space has a zero eigenvalue embedded in the neutral spectrum, i.e., there is no spectral gap. To resolve this issue, use is made of an exponentially weighted $$L^2$$ norm so that the essential spectrum is strictly shifted into the left-half plane, and this is closely related to the fact that solitary waves exist in the super-ion-sonic regime. Furthermore, in a certain long-wavelength scaling, we show that the Evans function for the Euler–Poisson system converges to that for the Korteweg–de Vries (KdV) equation as an amplitude parameter tends to zero, from which we deduce that the origin is the only eigenvalue on its natural domain with algebraic multiplicity two. We also show that the solitary waves are spectrally stable in $$L^2$$ space. Moreover, we discuss (in)stability of large amplitude solitary waves.

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

We investigate the roles of symmetry and bulk-boundary correspondence in characterizing topological edge states in generalized Jackiw–Rebbi (JR) models. We show that time-reversal (T), charge-conjugation (C), parity (P), and discrete internal field rotation ($$Z_n$$ Z n ) symmetries protect and characterize the various types of edge states such as chiral and nonchiral solitons via bulk-boundary correspondence in the presence of the multiple vacua. As two representative models, we consider the JR model composed of a single fermion field having a complex mass and the generalized JR model with two massless but interacting fermion fields. The JR model shows nonchiral solitons with the $$Z_2$$ Z 2 rotation symmetry, whereas it shows chiral solitons with the broken $$Z_2$$ Z 2 rotation symmetry. In the generalized JR model, only nonchiral solitons can emerge with only $$Z_2$$ Z 2 rotation symmetry, whereas both chiral and nonchiral solitons can exist with enhanced $$Z_4$$ Z 4 rotation symmetry. Moreover, we find that the nonchiral solitons have C, P symmetries while the chiral solitons do not, which can be explained by the symmetry-invariant lines connecting degenerate vacua. Finally, we find the symmetry correspondence between multiply-degenerate global vacua and solitons such that T, C, P symmetries of a soliton inherit from global minima that are connected by the soliton, which provides a novel tool for the characterization of topological solitons.

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

In this paper, Lie classical approach is utilized for the symmetry reduction of time-fractional variable coefficient Caudrey–Dodd–Gibbon–Sawada–Kotera equation. The obtained symmetries and Erd $$\acute{e}$$ lyi–Kober fractional differential operator are used to reduce the original nonlinear partial differential equation into nonlinear ordinary differential equation. The generalized Noether operator and new conservation theorem are exploited to obtain conservation laws of the governing equation. The power series solution is also derived for the considered equation. The obtained power series solution is investigated for the convergence and the obtained power series solution is convergent.

Optical fiber communication has developed rapidly because of the needs of the information age. Here, the variable coefficients fifth-order nonlinear Schrodinger equation (NLS), which can be used to describe the transmission of femtosecond pulse in the optical fiber, is studied. By virtue of the Hirota method, we get the one-soliton and two-soliton solutions. Interactions between solitons are presented, and the soliton stability is discussed through adjusting the values of dispersion and nonlinear effects. Results may potentially be useful for optical communications such as all-optical switches or the study of soliton control.

The fractional derivative method is used to take wave dispersion into account in the wave equation when describing the propagation of gravitational soliton waves on the surface of deep water. This approach is similar to that used to obtain the Korteweg–de Vries equation for solitons on the surface of shallow water, where the dispersion term in the wave equation is the third derivative of the velocity. It provides an alternative to the well-known approach of Zakharov and others based on the model of the nonlinear Schrodinger equation. The obtained nonlinear integral equation can be solved numerically.

The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized (2 + 1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili (CHKP) equation, which contains first-order, second-order, and third-order waves solutions. At the critical point, the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one minimum value. For the case, the lump solution will be shown the bright-dark lump structure and for another case can be present the dark lump structure-two small peaks and one deep hole. Also, the interaction of lump with periodic waves and the interaction between lump and soliton can be obtained by introducing the Hirota forms. In the meanwhile, the cross-kink wave and periodic wave solutions can be gained by the Hirota operator. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values. We alternative offer that the determining method is general, impressive, outspoken, and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering.

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

The purpose of this article is to investigate almost Ricci solitons on para- contact manifolds. We demonstrate that a gradient almost Ricci soliton whose metric is para-sasakian turns into Einstein with a constant scalar curvature -2n(2n + 1). Next, we get a few relationships between almost Ricci solitons and Ricci solitons on a K-paracontact manifold and its generalizations. Finally, some ndings on para-contact manifolds and H-paracontact manifolds admitting an almost Ricci soliton with a potential vector field that is a pointwise collinear with the Reeb vector field are discovered.&nbsp

In this paper, a novel reduction approach is reported for a physically meaningful nonlocal nonlinear Schrodinger (NLS) equation of reverse-time type to obtain its N-soliton solutions. Firstly, single-soliton solutions of the nonlocal NLS equation are obtained by reducing those of the local NLS equation. Secondly, inspired by the form of single-soliton solutions, N-soliton representations of the nonlocal NLS equation are conjectured and then verified via an algebraic proof. Thirdly, to demonstrate the features of the soliton solutions, some special soliton dynamics are theoretically explored and graphically illustrated. The reduction approach proposed in this paper has the merit that it is purely algebraic which does not need to perform complicated spectral analysis of the corresponding Lax pair.

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger (NLS) hierarchies associated with higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless inverse scattering transforms, is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

In this paper, a generalized Darboux transformation of multicomponent nonlinear Schrodinger (NLS) equations is constructed. Nth-order rogue wave solutions of the discussed multicomponent NLS equations are obtained by the resulting generalized Darboux transformation. As two illustrative examples, different kinds of solutions of three-component and six-component NLS equations are obtained, which include rogue wave solutions and interaction solutions.

In this paper, the soliton solutions which indicate long wave parallel to the magnetic fields of Kaup-Newell models are argued via described method. Modified extended Simple equation is suggested to explore the novel solitons and other wave structures of two different types of Kaup-Nawell equations, which have never been constructed before. As a consequence, bright-dark solitons, singular solitons, multi-wave solitons, breather type wave of strange structures and other waves solutions of two Kaup-Newell (K-N) Schrodinger equations are achieved in different form. The obtained novel solitons and other exact wave solutions have key applications in engineering and applied physics. Novel wave structures of solitons are explained graphically by providing suitable values to parameters that help for understanding the physical phenomena of these models. The constructed solutions are evaluated with available results in the literature. This technique can be productively utilized to more equations that occur in mathematical physics.

In this paper, we study some properties of multi-solitons for the non-linear Schrodinger equations in Rd with general non-linearities. Multi-solitons have already been constructed in H1(Rd) in pape...

On the basis of a $$3\times 3$$ matrix spectral problem, a second-order three-wave interaction system is proposed. A one-fold Darboux transformation is derived by using the gauge transformation between the $$3\times 3$$ matrix spectral problems. The compact determinant form of the N-fold Darboux transformation is obtained by iterating the onefold Darboux transformation and solving a complex linear algebraic system. Furthermore, the N-fold Darboux transformation and Taylor expansion are used to construct multifold generalized Darboux transforms of the second-order three-wave interaction system. As applications, the solutions of the interaction between a dark-bright soliton and a rogue wave (RW), the solutions of the interaction between two dark-bright solitons and a second-order rogue wave, the solutions of the eye-shaped rogue wave and triangle rogue wave are obtained.

Motivated by a recent development in the derivation of soliton solutions for initial-boundary value problems via the dressing method, we reconsider solutions to the nonlinear Schrödinger (NLS) equation on two half-lines connected via integrable defect conditions. The dressing method to construct soliton solutions is applied, while preserving the spectral boundary constraint with a time-dependent defect matrix. Moreover, N-soliton solutions are explicitly constructed and it is proven that the solitons are transmitted through the defect independently of one another.

The main scope of this paper is to examine the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on the space curve in E³. The differential geometric properties the soliton surface accociated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures k_{n}, k_{g} and τ_{g.}. Then, we give a different proof of that the s- parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame.

We consider a model that describes electromigration in nanoconductors known as surface electromigration (SEM) equation. Our purpose here is to establish local well-posedness for the associated initial value problem in Sobolev spaces from two different points of view. In the first one, we study the pure Cauchy problem and establish local well-posedness in $H^s(\mathbb{R}^2)$, $s>1/2$. In the second one, we study the Cauchy problem on the background of a Korteweg-de Vries solitary traveling wave in a less regular space. To obtain our results we make use of the smoothing properties of solutions for the linear problem corresponding to the Zakharov-Kuznetsov equation for the latter problem. For the former problem we use bilinear estimates in Fourier restriction spaces established by Molinet and Pilod.

Novel soliton structures are constructed for the Fokas–Lenells equation. In so doing, and after discussing the stability of continuous waves, a multiple scales based perturbation theory is used to reduce the equation to a Korteweg–de Vries system whose single soliton solution gives rise to intricate (and rather unexpected) solutions to the original system. Both the focusing and defocusing equations are considered and it is found that dark solitons may exist in both cases while in the focusing case antidark solitons are also possible. These findings are quite surprising as the relative nonlinear Schrödinger equation does not exhibit these solutions. So far, similar abundance of solutions has only been observed in relative coupled systems.

We study geometric aspects of the imaginary Schwarzian \(S_2f\) for curves in 3-space, as introduced by Ahlfors in [1]. We show that \(S_2f\) points in the direction from the center of the osculating sphere to the point of contact with the curve. We also establish an important law of transformation of \(S_2f\) under Mobius transformations. We finally study questions of existence and uniqueness up to Mobius transformations of curves with given real and imaginary Schwarzians. We show that curves with the same generic imaginary Schwarzian are equal provided they agree to second order at one point, while prescribing in addition the real Schwarzian becomes an overdetermined problem.

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

Under investigation in this paper is a discrete modified exponential Toda lattice equation which may describe vibration of particles in a lattice. Firstly, we construct an integrable lattice hierarchy associated with this equation from a $$2\times 2$$ matrix spectral problem, and some related integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws of this hierarchy are discussed. Secondly, we present the discrete generalized $$(m, N-m)$$ -fold Darboux transformation of the modified exponential Toda lattice equation on the basis of its known Lax representation. As applications of the obtained discrete generalized Darboux transformation, multi-kink-soliton solutions on an exponential surface background and an inclined plane background are obtained when $$m=2N$$ , the discrete rational and semi-rational solutions are derived when $$m=1$$ , and the mixed solutions of usual soliton solutions and rational solutions are given when $$m=2$$ . Based on the asymptotic and graphic analysis, soliton elastic interaction phenomena and limit states related to rational solutions are discussed and analyzed. Furthermore, some mathematical features of rational solutions are summarized. Finally, numerical simulations are used to explore the dynamical behaviors of such soliton solutions which show the soliton evolutions are robust against a small noise. These results and properties given in this paper might be useful for understanding nonlinear lattice dynamics.

The object of the present paper is to study some properties of Kenmotsu manifold whose metric is conformal $\eta$-Einstein soliton. We have studied some certain properties of Kenmotsu manifold admitting conformal $\eta$-Einstein soliton. We have also constructed a 3-dimensional Kenmotsu manifold satisfying conformal $\eta$-Einstein soliton., Comment: 10 pages

In this paper by using Hirota’s method, the one and two soliton solutions of nonlinear Schr¨odinger equation with self-consistent source are studied. We have shown the evolution of the one and two soliton solutions in detail by using graphics.

This article studies a (2+1)–dimensional first extended Calogero-Bogoyavlenskii-Schiff equation, which was recently introduced in the literature. We derive Lie symmetries of this equation and then use them to perform symmetry reductions. Using translation symmetries, a fourth-order ordinary differential equation is obtained which is then solved with the aid of Kudryashov and $ (G'/G)- $expansion techniques to construct closed-form solutions. Besides, we depict the solutions with the appropriate graphical representations. Moreover, conserved vectors of this equation are computed by engaging the multiplier approach as well as Noether's theorem.

Based on the degenerate Darboux transformation, the n-positon solution of the higher-order Chen–Lee–Liu (HOCLL) equation are obtained by the special limit $$\lambda _{j}\rightarrow \lambda _{1}$$ taking from the corresponding n-soliton solution, and using the higher-order Taylor expansion. Using the method of the modulus square decomposition, n-positon is decomposed into n single soliton solutions. The dynamic properties of smooth positon of the HOCLL equation are discussed in detail, and the corresponding trajectory, approximate trajectory and “phase shift” are obtained. In addition, the mixed solutions of soliton and positon are discussed, and the corresponding three-dimensional map are given.

The interaction of an oblique line soliton with a one-dimensional dynamic mean flow is analyzed using the Kadomtsev-Petviashvili II (KPII) equation. Building upon previous studies that examined the transmission or trapping of a soliton by a slowly varying rarefaction or oscillatory dispersive shock wave in one space and one time dimension, this paper allows for the incident soliton to approach the changing mean flow at a nonzero oblique angle. By deriving invariant quantities of the soliton-mean flow modulation equations$-$a system of three (1+1)-dimensional quasilinear, hyperbolic equations for the soliton and mean flow parameters$-$and positing the initial configuration as a Riemann problem in the modulation variables, it is possible to derive quantitative predictions regarding the evolution of the line soliton within the mean flow. It is found that the interaction between an oblique soliton and a changing mean flow leads to several novel features not observed in the (1+1)-dimensional reduced problem. Many of these interesting dynamics arise from the unique structure of the modulation equations that are nonstrictly hyperbolic, including a well-defined multivalued solution interpreted as a solution of the (2+1)-dimensional soliton-mean modulation equations, in which the soliton interacts with the mean flow and then wraps around to interact with it again. Finally, it is shown that the oblique interactions between solitons and dispersive shock wave solutions for the mean flow give rise to all three possible types of 2-soliton solutions of the KPII equation. The analytical findings are quantitatively supported by direct numerical simulations., 27 pages, 19 figures

A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. The new identity for theta-functions of curves with holomorphic involution having fixed points is obtained., 44 pages, no figures

In this paper, outcomes of the study on the bilinear forms and vector bright solitons for a coupled nonlinear Schrodinger system with variable coefficients are presented, which describes the simult...

The exact solutions to the Riemann problem for an inhomogeneous extended Chaplygin gas equations with friction are constructed explicitly. Compared to the homogeneous system, the Riemann solutions for the inhomogeneous one are no longer self-similar. Then, as the two exponents vanish wholly or partly, two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock waves are identified and investigated. It is rigorously proved that as the pressure tends to a constant, the Riemann solutions of the inhomogeneous extended Chaplygin gas equations converge to those of the zero-pressure flow with a body force, while as the pressure approaches some special generalized Chaplygin gas, the Riemann solutions of the inhomogeneous extended Chaplygin gas equations tend to those of the generalized Chaplygin gas equations with the same source term.

In this paper, we consider the long wavelength limit for the Euler-Poisson system arising in plasma including three species. It is demonstrated that when the plasma has critical densities, the modified Korteweg-de Vries (mKdV) equation can be derived, under the classical Gardner-Morikawa transform e 1 / 2 ( x − V t ) → x , e 3 / 2 t → t as e → 0 , with the velocity V depending on the densities. We estimate the error between the mKdV equation and the Euler-Poisson system in Sobolev spaces. The mKdV dynamics can be seen at time interval of order O ( e − 1 ) .

Under the framework of the complex column-vector loop algebra Cp, we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme. As applications of the scheme, we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model. The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems, including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell. Specially, we obtain the famous Fokker-Plank equation and its generalized form, which has extensive applications in the stochastic dynamic systems. Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).

We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the last., Comment: 13 p; v2: references to the works of Miller and Ogievetsky & Schechtman were added

This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and fractional cubic nonlinear Schrodinger (fNLS) equations, with the dispersions |D|α∂x and |D|α+1, respectively. We p...

We consider some direct and inverse problems associated with the vibration of an elastic conductive body governed by the Lame and Maxwell equations coupled through the nonlinear magnetoelastic effe...

The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known potentials of the Dirac operator and this algorithm is said to be Moutard transformations [5]. Firstly, the geometric meaning of these transformations which found in [6], [7], gives us the definition of the inversion of the minimal surface, further after finding the exact solutions of the mVN equation, we can represent the inverted surfaces. And these representations of the new potential determine the soliton deformation [8], [9]. In 2014, blowing-up solutions to the mVN equation were obtained using a rigid translation of the initial Enneper surface in [6]. Further results were obtained for the second-order Enneper surface [10]. Now the soliton deformation of an inverted catenoid is found by smooth translation along the second coordinate axis. In this paper, in order to determine catenoid inversions, it is proposed to find holomorphic objects as Gauss maps and height differential [11]; the soliton deformation of the inverted catenoid is obtained; particular solution of modified Karteweg-de Vries (KdV) equation is found that give some representation of KdV surface [12],[13].

At present, the question of studying multidimensional nonlinear integrable equations in the framework of the theory of solitons is very interesting to foreign and Kazakh scientists. Many physical phenomena that occur in nature can be described by nonlinearly integrated equations. Finding specific solutions to such equations plays an important role in studying the dynamics of phenomena occurring in various scientific and engineering fields, such as solid state physics, fluid mechanics, plasma physics and nonlinear optics. There are several methods for obtaining real and soliton, soliton-like solutions of such equations: the inverse scattering method, the Hirota’s bilinear method, Darboux transformation methods, the tanh-coth and the sine-cosine methods. In our work, we studied the two-dimensional Hirota equation, which is a modified nonlinear Schrödinger equation. The nonlinear Hirota equation is one of the integrating equations and the Hirota system is used in the field of study of optical fiber systems, physics, telecommunications and other engineering fields to describe many nonlinear phenomena. To date, the first, second, and n-order Darboux transformations have been developed for the two- dimensional system of Hirota equations, and the soliton, rogue wave solutions have been determined by various methods. In this article, we consider the two-dimensional nonlinear Hirota equations. Using the Lax pair and Darboux transformation we obtained the first and the second multi-line soliton solutions for this equation and provided graphical representation.

The study of integrable equations is one of the most important aspects of modern mathematical and theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion. Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the equation compensates for the dispersion term. In addition to integrable nonlinear differential equations, there is also an important class of integrable partial differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless (quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of known integrable equations. In recent years, the study of dispersionless systems has become of great importance, since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations. In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell- Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB using the obtained Lax representation.

The (2+1)-dimensional generalized coupled nonlinear Schrodinger equation with the four-wave mixing (FWM) term is studied in this paper, which describes the optical solitons in a birefringent fiber. By virtue of the Hirota method, the one- and two-soliton solutions are derived. On the basis of solutions obtained, we discuss how the values of the FWM and some free parameters affect the solitons’ peformance. The FWM parameter can help to control the amplitude of the solitons. Meanwhile, by setting the values of certain free parameters, we can control the solitons’ propagation direction and speed, and reduce the interactions between them as well. In addition, the energy transfer of solitons during elastic collision and separation is also discussed. The conclusions here may be useful in improving the communication quality in multi-mode fibers.

Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of $I$-method, we give its global well-posedness in lower regularity Sobolev spaces, and then obtain the asymptotic behavior in these spaces with weights.

We deduce a generalized hydrodynamic limit for the box-ball system, which explains how the densities of solitons of different sizes evolve asymptotically under Euler space-time scaling. To describe the limiting soliton flow, we introduce a continuous state-space analogue of the soliton decomposition of Ferrari, Nguyen, Rolla and Wang (cf. the original work of Takahashi and Satsuma), namely we relate the densities of solitons of given sizes in space to corresponding densities on a scale of 'effective distances', where the dynamics are linear. For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the 'effective speeds' of solitons locally.

Most of the existing research on gyrostat systems uses the Melnikov method to analyze chaotic dynamic characteristics. The mechanism of chaotic motion from the energy perspective of gyrostat systems has rarely been analyzed. The gyrostat system with positive damping disturbance is modeled in this paper. Casimir energy and power are given. The mechanism of the orbital expansion and contraction of the gyrostat is revealed through the sign change of Casimir power and the ellipsoid of Casimir energy. The bifurcations of the Casimir power corresponding to the state bifurcation and Lyapunov exponent spectrum reflect the evolution and transition of gyrostat's dynamics. The energy-level leap expressed by the variance of Casimir energy is found to determine the bifurcation of dynamics for the first time. The coexistence of the gyrostat is explained using Casimir power and local stability of equilibrium point. The COMSOL finite element simulation platform is built to study the state changes of the displacement orbit of the frame gyrostat model. A comparison is performed for the displacement results conducted by the COMSOL and the numerical simulation for the gyrostat system kinematic model. The different displacement states of the gyrostat system under free motion and in the presence of damping disturbance and external torques are obtained.

For a coupled nonlinear Schrodinger system with the negative coherent coupling, which describes two orthogonally polarized pulses in a weakly birefringent fibre, we construct the Darboux dressing t...