Nizhnik Novikov Veselov Research Articles

Characteristics of the breathers, rogue waves and soliton waves in a (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov equation

In this work, a (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) equation, which can be reduced to several integrable equations, is under investigation. By virtue of Bell’s polynomials, an effective and straightforward way is presented to succinctly construct its two bilinear forms. Furthermore, based on the bilinear formalism and the extended homoclinic test, the breather wave solution, rogue-wave solution and solitary-wave solution of the equation are well constructed. The results can be used to enrich the dynamical behavior of the (2 + 1)-dimensional nonlinear wave fields.

Conservation laws, solitons, breather and rogue waves for the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium

In this paper, we consider the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium, which is the isotropic Lax integrable extension of the Korteweg-de Vries equation. Infinitely-many conservation laws are constructed. N-soliton solutions in terms of the Wronskian are obtained via the Hirota method. Velocity and shape of the soliton can be influenced by those variable coefficients, while the amplitude of the soliton can not be affected. Collision between the two solitons is shown to be elastic. Breather wave solutions are constructed via the trilinear method. Such phenomena as the deceleration and compression of the breather waves are studied graphically. Rogue wave solutions are derived when the periods of the breather wave solutions go to the infinity. Separated and united composite rogue waves are discussed graphically.

New solutions for conformable fractional Nizhnik–Novikov–Veselov system via $$G'/G$$ G ′ / G expansion method and homotopy analysis methods

The main purpose of this paper is to find the exact and approximate analytical solution of Nizhnik–Novikov–Veselov system which may be considered as a model for an incompressible fluid with newly defined conformable derivative by using \(G'/G\) expansion method and homotopy analysis method (HAM) respectively. Authors used conformable derivative because of its applicability and lucidity. It is known that, the NNV system of equations is an isotropic Lax integrable extension of the well-known KdV equation and has physical significance. Also, NNV system of equations can be derived from the inner parameter-dependent symmetry constraint of the KP equation. Then the exact solutions obtained by using \(G'/G\) expansion method are compared with the approximate analytical solutions attained by employing HAM.

Lumps and rogue waves of generalized Nizhnik–Novikov–Veselov equation

We investigate the generalized $$(2+1)$$ Nizhnik–Novikov–Veselov equation and construct its linear eigenvalue problem in the coordinate space from the results of singularity structure analysis thereby dispelling the notion of weak Lax pair. We then exploit the Lax pair employing Darboux transformation and generate lumps and rogue waves. The dynamics of lumps and rogue waves is then investigated.

Analysis of the generalized (2+1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium

Korteweg-de Vries (KdV)-type equations are seen to describe the shallow-water waves, lattice structures and ion-acoustic waves in plasmas. Hereby, we consider an extension of the KdV-type equations called the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium. Via the Hirota bilinear method and symbolic computation, we derive the bilinear forms, N-soliton solutions and Bäcklund transformation. Effects of the first- and higher-order dispersion terms are investigated. Soliton evolution and interaction are graphically presented and analyzed: Both the propagation velocity and direction of the soliton change when the dispersion terms are time-dependent; The interactions between/among the solitons are elastic, independent of the forms of the coefficients in the equations.

Lump soliton, mixed lump stripe and periodic lump solutions of a (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

The lump soliton solutions of a (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation are obtained by making use of its bilinear form. We discuss the conditions to guarantee the analyticity, positiveness and localization of lump solutions. The solutions of interaction between a lump and a stripe are presented. It is proved that the interaction between the two solitary waves is non-elastic. The three-wave method is employed to investigate the periodic lump solutions. Figures are presented to illustrate the dynamical features of these solutions.

Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations

In this paper, we present a generalized unified method for finding multiwave solutions of the time-fractional (2+1)-dimensional Nizhnik–Novikov–Veselov equations. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation. Multiauxiliary equations have been introduced in this method to obtain not only multisoliton solutions but also multiperiodic or multielliptic solutions. It is shown that the considered method is very effective and convenient for solving wide classes of nonlinear partial differential equations of fractional order.

A new analytical method for seeking traveling wave solutions of space–time fractional partial differential equations arising in mathematical physics

In this paper, based a new fractional sub-equation and the properties of the modified Riemann–Liouville fractional derivative, we propose a new analytical method named improved fractional (DαG/G) method for seeking traveling wave solutions of space–time fractional partial differential equations, which can be seen as the fractional version of the improved (G′/G) method, and is the improvement of the common fractional (DαG/G) method. Due to given traveling wave transformation, one certain space–time fractional partial differential equation can be converted into another fractional ordinary differential equation with respect to one new variable. For demonstrating the validity of this method, we apply it to seek exact traveling wave solutions for the (2+1)-dimensional space–time fractional Nizhnik–Novikov–Veselov System and the space–time fractional KP-BBM equation. With the aid of the mathematical software, some new exact traveling wave solutions including solitary wave solutions, periodic wave solutions and solutions with other forms for them are successfully found.

Nonlocal Symmetry, CRE Solvability and Exact Interaction Solutions of the Asymmetric Nizhnik–Novikov–Veselov System

Abstract Applying the truncated Painlevé expansion to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system, some Bäcklund transformations (BTs) including auto BT and non-auto BT are obtained. The auto BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion and the related nonlocal symmetry group is presented with the help of the localization procedure. Further, it is shown that the ANNV system has a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, some complex interaction solutions between soliton and arbitrary other seed waves of the ANNV system are readily constructed, such as bight-dark soliton solution, dark-dark soliton solution, soliton-cnoidal interaction solutions, solitoff solutions and so on.

Note on the Equivalence of Variable Separation Solutions Based On the Improved tanh-Function Method

Abstract The equivalence of variable separation solutions based on the improved tanh-function method (ITM) for nonlinear models is illustrated. As an example, we restudy the (2+1)-dimensional generalised Nizhnik–Novikov–Veselov system via the ITM. Based on the radical sign-combined ansatz, five types of variable separation solutions are obtained. By careful analysis, we prove that these seemingly independent variable separation solutions actually depend on each other.

Dynamic behaviors of bright and dark rogue waves for the (2+1) dimensional Nizhnik–Novikov–Veselov equation

This paper investigates the real (2+1)-dimensional Nizhnik–Novikov–Veselov equation. Using the Hirota trilinear form and the Taylor expansion method, the rational rogue wave is constructed. The complicated structures of rogue wave—including bright rogue wave, four-lump type rogue wave, and dark rogue wave—are presented. The existence conditions of bright and dark rogue waves are given. The dynamic behaviors of rogue waves are discussed mathematically and graphically. These results demonstrate the diversity of the structures of the rational rogue wave to the real system. It is hoped that these results may be useful for explaining some related nonlinear phenomena.

Consistent Riccati Expansion for Integrable Systems

A consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a Riccati equation. A system having a CRE is then defined to be CRE solvable. The CRE solvability is demonstrated quite universal for various integrable systems including the Korteweg–de Vries, Kadomtsev–Petviashvili, Ablowitz–Kaup–Newell–Segur (and then nonlinear Schrödinger), sine‐Gordon, Sawada–Kotera, Kaup–Kupershmidt, modified asymmetric Nizhnik–Novikov–Veselov, Broer–Kaup, dispersive water wave, and Burgers systems. In addition, it is revealed that many CRE solvable systems share a similar determining equation describing the interactions between a soliton and a cnoidal wave. They have a common nonlocal symmetry expression and they possess a formally universal once Bäcklund transformation.

Bifurcations of traveling wave solutions for the (2 + 1)-dimensional generalized asymmetric Nizhnik–Novikov–Veselov equation

By using the bifurcation theory of planar dynamical systems to the (2+1)-dimensional generalized asymmetric Nizhnik–Novikov–Veselov equation, the existence for solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of these solutions mentioned are given. Furthermore, some exact explicit parametric expressions of these bounded traveling waves are obtained.

Conservation laws, Lie symmetry and Painlevé analysis of the variable coefficients NNV equation

In this paper, with symbolic computation, Lie symmetry analysis, Painlevé test, conservation laws and similarity solutions for the generalized (2+1)-dimensional variable coefficients Nizhnik–Novikov–Veselov (VCNNV) equation are studied. Firstly, we derive the group classifications and the corresponding symmetry reductions via Lie group method. Then some new variable separation solutions with Lie symmetry properties are discussed. And integrable conditions of such system are determined via the Painlevé analysis. At last, some new infinite time-dependent conservation laws are derived, base on the “new conservation theorem” proved by Ibragimov.

About linear superpositions of special exact solutions of Nizhnik-Veselov-Novikov equation

General scheme for calculations via Zakharov and Manakov ∂̄-dressing method of exact solutions, nonstationary and stationary, of Nizhnik-Veselov-Novikov (NVN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u(n), n = 1,..., N is presented. Simple nonlinear superpositions are given up to a constant by the sums of solutions u(n) and calculated by ∂̄-dressing of the first auxiliary linear problem with nonzero asymptotic values of potential at infinity. It is remarkable that in the zero limit of asymptotic values of potential simple nonlinear superpositions convert in to linear ones in the form of the sums of special solutions u(n). It is shown that the sums u = u(kl) + ...+ u(km), 1 ≤ k1 < k2 < ... < km ≤ N of arbitrary subsets of these N solutions are also exact solutions of NVN equation. The obtained results are illustrated in detail by hyperbolic version of NVN equation, i. e. by NVN-II equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions.

Elastic and Inelastic Interaction Behaviours for the (2+1)-Dimensional Nizhnik–Novikov–Veselov Equation in Water Waves

A modified mapping method and new ansätz form are used to derive three families of variable separation solutions with two arbitrary functions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation in water waves. By selecting appropriate functions in the variable separation solution, we discuss interaction behaviours among dromion-pair and dromion-like peakon-pair and dromionlike semifoldon-pair. The analysis results exhibit that the interaction behaviours between dromionpair and dromion-like peakon-pair, dromion-pair and semifoldon-pair, dromion-like peakon-pair and semifoldon-pair are all incomplete elastic, and there exists a phase shift. The interaction behaviour between two dromion-like semifoldon-pairs is completely elastic, and no phase shift appears after interaction. Moreover, during the interactions between dromion-pair and semifoldon-pair, dromionlike peakon-pair and semifoldon-pair, and between two dromion-like semifoldon-pairs, there all exist a multi-valued semifoldon-pair.

New Similarity Reduction Solutions for the (2+1)-Dimensional Nizhnik—Novikov—Veselov Equation

In this paper, some new formal similarity reduction solutions for the (2+1)-dimensional Nizhnik-Novikov-Veselov equation are derived. Firstly, we derive the similarity reduction of the NNV equation with the optimal system of the admitted one-dimensional subalgebras. Secondly, by analyzing the reduced equation, three types of similarity solutions are derived, such as multi-soliton like solutions, variable separations solutions, and KdV type solutions.

Localized Structures Constructed by Multi-Valued Functions in the (2+1)-dimensional Generalized Nizhnik–Novikov–Veselov Equation

Localized Structures Constructed by Multi-Valued Functions in the (2+1)-dimensional Generalized Nizhnik–Novikov–Veselov Equation

Bäcklund transformation and Wronskian solitons for the [formula omitted]-dimensional Nizhnik–Novikov–Veselov equations

Korteweg–de Vries-type equations are seen to describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, an isotropic extension of which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equations. Hereby, based on the Hirota bilinear method and symbolic computation, we derive the bilinear form and Bäcklund transformation for such an extension. N-soliton solutions in the Wronskian form are given, and it can be verified that the Bäcklund transformation can connect the (N−1)- and N-soliton solutions. Solitonic propagation and collision are discussed: the larger-amplitude soliton moves faster and then overtakes the smaller one. After the collisions, the solitons keep their original shapes and velocities invariant except for the phase shift. Collisions among the two and three solitons are all elastic.

Double traveling wave solutions for some nonlinear partial differential equations in mathematical Physics

In this article, we construct the double soliton exact solutions for some nonlinear partial differential equations in mathematical physics via (2+1)-dimensional Painleve integrable Burgers equations, (2+1)-dimensional breaking soliton equations and (2+1)-dimensional Nizhnik-Novikov Veselov equations. We obtain many new types of complexiton soliton solution, that is, various combination of trigonometric periodic function and hyperbolic function, various combination of trigonometric periodic function and rational function, various combination of hyperbolic function and rational function solutions. Key words: Double soliton solution, the traveling wave solution, the (2+1)-dimensional Painleve integrable Burgers equations, the (2+1)-dimensional breaking soliton equations and the (2+1)-dimensional Nizhnik-Novikov Veselov equations.