Novikov Veselov Research Articles

Dynamics of mixed lump-soliton for an extended (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation

A new (2+1)-dimensional higher-order extended asymmetric Nizhnik–Novikov–Veselov (eANNV) equation is proposed by introducing the additional bilinear terms to the usual ANNV equation. Based on the independent transformation, the bilinear form of the eANNV equation is constructed. The lump wave is guaranteed by introducing a positive constant term in the quadratic function. Meanwhile, different class solutions of the eANNV equation are obtained by mixing the quadratic function with the exponential functions. For the interaction between the lump wave and one-soliton, the energy of the lump wave and one-soliton can transfer to each other at different times. The interaction between a lump and two-soliton can be obtained only by eliminating the sixth-order bilinear term. The dynamics of these solutions are illustrated by selecting the specific parameters in three-dimensional, contour and density plots.

Abundant traveling wave and numerical solutions for Novikov-Veselov system with their stability and accuracy

Solutions such as symmetric, bright soliton and periodic solutions play a prominent role in the field of differential equations, and they can be used to investigate several phenomena in nonlinear sciences. Some waves such as ion and magneto-sound waves in plasma are investigated by using some partial differential equations (PDEs) such as Novikov-Veselov (NV) equations. In this work, the improved -expansion approach is utilized to extract numerous soliton solutions for NV system. Hamiltonian system is invoked to analyze the stability of some solutions. The finite difference approach is successfully applied to achieve the numerical simulations of the proposed equations. We also introduce the stability and the accuracy of the numerical scheme. In order to validate the correctness of the accomplished results, we compare the exact solutions with the numerical solutions analytically and graphically. The presented techniques are very convenient and adequate and can be employed to other types of nonlinear evolution equations.

Nonlinear superposition between lump waves and other waves of the (2 $$+$$ 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Nonlinear superposition between lump waves and other waves of the (2 $$+$$ 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

D’Alembert wave and soliton molecule of the generalized Nizhnik–Novikov–Veselov equation

Traveling wave solution is one of the effective methods for solving nonlinear partial differential equations. D’Alembert solution is a special kind of traveling wave solution. There have been many studies about D’Alembert solution. In this paper, we will solve D’Alembert-type wave solutions for (2+1)-dimensional generalized Nizhnik–Novikov–Veselov equation. Based on the Hirota bilinear transformation and velocity resonance mechanism, the states of soliton molecules composed of two solitons, three solitons and four solitons are studied. It is concluded that D’Alembert-type wave is closely related to soliton molecules.

ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

A Hausdorff fractal Nizhnik-Novikov-Veselov model arising in the incompressible fluid

PurposeNizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an incompressible fluid. The purpose of this paper is to present a fractal model of the NNVS based on the Hausdorff fractal derivative fundamental concept.Design/methodology/approachA two-scale transformation is used to convert the proposed fractal model into regular NNVS. The variational strategy of well-known Chinese scientist Prof. Ji Huan He is used to generate bright and exponential soliton solutions for the proposed fractal system.FindingsThe NNV fractal model and its variational principle are introduced in this paper. Solitons are created with a variety of restriction interactions that must all be applied equally. Finally, the three-dimensional diagrams are displayed using an appropriate range of physical parameters. The results of the solitary solutions demonstrated that the suggested method is very accurate and effective. The proposed methodology is extremely useful and nearly preferable for use in such problems.Practical implicationsThe research study of the soliton theory has already played a pioneering role in modern nonlinear science. It is widely used in many natural sciences, including communication, biology, chemistry and mathematics, as well as almost all branches of physics, including nonlinear optics, plasma physics, fluid dynamics, condensed matter physics and field theory, among others. As a result, while constructing possible soliton solutions to a nonlinear NNV model arising from the field of an incompressible fluid is a popular topic, solving nonlinear fluid mechanics problems is significantly more difficult than solving linear ones.Originality/valueTo the best of the authors’ knowledge, for the first time in the literature, this study presents Prof. Ji Huan He's variational algorithm for finding and studying solitary solutions of the fractal NNV model. The reported solutions are novel and present a valuable addition to the literature in soliton theory.

Resonance [formula omitted]-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

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

Soliton molecules in the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

The [Formula: see text]-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for [Formula: see text]. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution [Formula: see text] are constructed for [Formula: see text]. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable [Formula: see text] for this equation are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

The influence of multiplicative noise on the stochastic exact solutions of the Nizhnik–Novikov–Veselov system

In this paper we consider the stochastic Nizhnik–Novikov–Veselov system, which is forced in the Itô sense by multiplicative noise. To get a new trigonometric and hyperbolic stochastic solutions, we use the tanh–coth method and sine–cosine method. In addition, we demonstrate the effect of multiplicative noise on the exact solutions of the stochastic Nizhnik–Novikov–Veselov system

The lump solutions for the (2+1)-dimensional Nizhnik–Novikov–Veselov equations

The exact solutions for the (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equations are investigated via Maple software. According to the Hirota bilinear form of the NNV equations, lump solutions are mainly obtained by the limit method of the heteroclinic test functions with initial values. Furthermore, the interaction solutions of the lump and N-soliton solutions are researched with the aid of new test functions which consist of the quadratic polynomial functions and the exponential functions. Especially, the sufficient conditions for the existence of the interactions solutions for NNV equations are given.

On determining some exact wave solutions to the Nizhnik–Novikov–Veselov system via a rebuts technique

In this study, a significant model of mathematical physics used to describe real-world problems, namely the Nizhnik–Novikov–Veselov (NNV) system is studied and various versions of wave solutions are simultaneously explored. The NNV system is considered an essential model to the KdV hierarchy and is widely used to study the scalar nucleons associated with neutral scalar masons. Thus, the exact wave solutions are important to the associated researchers. To this aim, a robust approach named the generalized exponential rational function method (GERFM) is used to achieve this goal. With the aid of GERFM abundant new wave solutions are formally constructed. All generated solutions are constructed with distinct forms that are dependent on free parameters and determined by the self-form of the NNV system. Moreover, the free parameters play the main roles in the dispersive terms and the values of free parameters influence the physical properties of the traveling solitary and periodic waves to the NNV system directly. Finally, some 3D graphs of the generated solutions are provided to show the propagations of traveling waves. The performed results will have great potential applications to optical fibers, magneto-electrodynamics, the collision of heavy ions, and quantum mechanics.

Numerical soliton solutions of fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations in nonlinear optics

In this paper, time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations have been solved numerically utilizing the Kansa method, in which the multiquadrics are taken as radial basis function. To attain this, a numerical scheme based on finite difference and Kansa method has been proposed. In addition, the soliton solutions have been obtained by employing Kudryashov method and tanh method for comparison purpose with the obtained numerical solutions. The numerical examples are given to demonstrate the accuracy and applicability of the proposed method.

Using the improved $$\exp ( - \Phi (\xi ))$$ expansion method to find the soliton solutions of the nonlinear evolution equation

With the help of the symbolic calculation software Mathematica, the nonlinear coupled Schrodinger system, (2 + 1)-dimensional dispersive long-wave system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system are solved by using the improved $$\exp ( - \Phi (\xi ))$$ expansion method. Diverse new soliton solutions are obtained, and the structure and properties of soliton solutions are studied. We derive seven sets of exact solutions under different conditions for the nonlinear coupled Schrodinger system and the (2 + 1)-dimensional dispersive long wave equation, respectively, and for the (2 + 1)-dimensional Nizhnik–Novikov–Veselov system, we find the inelastic interaction solitary waves and fussion soliton.

Linear and nonlinear effects analysis on wave profiles in optics and quantum physics

In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters.

Rational and semi-rational solutions to the asymmetric Nizhnik–Novikov–Veselov system

In this paper, we use the binary Darboux transformation technique to derive an uniform mathematical expression of all kinds of solutions to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. For the same seeding solution, a family of eigenfunctions associated with the same eigenvalue is obtained, which is used to construct rational and semi-rational solutions. Interestingly, there exists a category of localized rational solutions that show nontrivial interaction scenarios, namely the pulses undergo a scattering process after the head-on collision. The semi-rational solutions are characterized by two generic evolution scenarios: fission and fusion processes. We also find a subclass of dark rogue waves, namely trains of line solitons that evolve to significant strongly localized transient waves.

D’Alembert wave and soliton molecule of the modified Nizhnik–Novikov–Veselov equation

The wave motion equation is one of the fundamental equations to describe vibrations of continuous systems. The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential equations. The study of the D’Alembert wave deserves deep consideration in nonlinear equations. In this paper, the D’Alembert-type wave of the (2 + 1)-dimensional modified Nizhnik–Novikov–Veselov (mNNV) equation is derived by the Ansatze method. The Hirota bilinear form of the mNNV equation is constructed by introducing the dependent variable transformation. The multi-soliton solution is obtained by solving the corresponding bilinear form. By combining the velocity resonance mechanism, a three-soliton molecule, the interaction between a soliton molecule and one soliton, and the interaction between a soliton–solitoff molecule and one soliton of the mNNV equation are obtained. The dynamics of these solutions are shown by selecting the appropriate parameters. These phenomena for the mNNV equation have not yet been given via other methods.

The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

N‐lump and interaction solutions of localized waves to the (2 + 1)‐dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid

The present article deals with M‐soliton solution and N‐soliton solution of the (2 + 1)‐dimensional asymmetrical Nizhnik–Novikov–Veselov equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump, and their interactions, which have been investigated by the approach of the long‐wave limit. Mainly, by choosing the specific parameter constraints in the M‐soliton and N‐soliton solutions, all cases of the one breather or one lump can be captured from the two, three, four, and five solitons. In addition, the performances of the mentioned technique, namely, the Hirota bilinear technique, are substantially powerful and absolutely reliable to search for new explicit solutions of nonlinear models. Meanwhile, the obtained solutions are extended with numerical simulation to analyze graphically, which results in localized waves and their interaction from the two‐, three‐, four‐, and five‐soliton solutions profiles. They will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.

Dynamical interactions between higher-order rogue waves and various forms of n-soliton solutions (n → ∞) of the (2+1)-dimensional ANNV equation

We present new lemmas, theorem and corollaries to construct interactions among higher-order rogue waves, n-periodic waves and n-solitons solutions (n → ∞) to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation. Several examples for theories are given by choosing definite interactions of the wave solutions for the model. In particular, we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave, a rogue and a cross-bright bell wave, a rogue and a one-, two-, three-, four-periodic wave. In addition, we also present multi-types interactions between a rogue and a periodic cross-bright bell wave, a rogue and a periodic cross-bright-bark bell wave. Finally, we physically explain such interaction solutions of the model in the 3D and density plots.

Abundant structures of waves in plasma transitional layer sheath

In this paper multi-wave solutions representing coupled self-similar-traveling wave of the Nizhnik Novikov Veselov equation (NNVE) in a non-uniform medium, are derived. The solutions of (NNVE) describe waves in plasma transitional layer sheath and reveal that they exhibit symmetrically profile about x=0 and y=0. This behavior may be due to a fact that medium acts as a barrier, with high refractive index at the center of the medium. Different waves structures waves in the form of water fall, W-, V- and S- soliton-waves shapes are shown. Further the solutions exhibit undulated soliton waves and waves islands. Here, the proposed approach to find the solutions uses the extended (and generalized) unified methods.