Nizhnik Novikov Veselov System Research Articles

New periodic and soliton wave solutions for the generalized Zakharov system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system

In this paper, the Exp-function method is used to obtain generalized solitonary solutions and periodic solutions of the Generalized Zakharov system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Folded Solitary Wave Excitations for (2+1)-Dimensional Nizhnik–Novikov–Veselov System

With the help of an improved mapping approach and a linear-variable-separation approach, a new family of exact solutions with arbitrary functions of the (2+1)-dimensional Nizhnik–Novikov–Veselov system (NNV) is derived. Based on the derived solutions and using some multi-valued functions, we find a few new folded solitary wave excitations for the (2+1)-dimensional NNV system.

New variable separation solutions of the [formula omitted]-dimensional asymmetric Nizhnik–Novikov–Veselov system

By means of the exp-function method, we obtain some new exact solutions of the Riccati equation. Based on the Riccati equation and its new exact solutions, we find new and more general variable separation solutions with two arbitrary functions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system. As some special examples, these new variable separation solutions can degenerate into some known solutions reported in open literature.

A re-study on the (2+1)-dimensional Nizhnik–Novikov–Veselov system by the generalized mapping method

The generalized mapping method is introduced to solve the (2+1)-dimensional Nizhnik–Novikov–Veselov system. New families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions are derived. Based on the periodic wave solution and by selecting appropriate functions, the evolutional behaviors of dromions and compactons on the background of the Jacobian elliptic wave are investigated.

Combined wave solutions of the -dimensional generalized Nizhnik–Novikov–Veselov system

Using symbolic and algebra computation, the extended tanh-function method (ETM) based on mapping method is further improved. The new variable separation solutions of the (2+1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system are derived. From the periodic wave solution and by selecting appropriate functions, the evolutional behaviors of compactons in the background of Jacobian elliptic wave and their interaction behaviors are investigated.

Exotic interactions between solitons of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov system

Starting from the extended tanh-function method (ETM) based on the mapping method, the variable separation solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semi-foldon, are investigated.

NEW TYPES OF INTERACTIONS BASED ON VARIABLE SEPARATION SOLUTIONS VIA THE GENERAL PROJECTIVE RICCATI EQUATION METHOD

In this paper, first, the general projective Riccati equation method (PREM) is applied to derive variable separation solutions of (2 + 1)-dimensional systems. By further studying, we find that these variable separation solutions obtained by PREM, which seem independent, actually depend on each other. A common formula with some arbitrary functions is obtained to describe suitable physical quantities for some (2 + 1)-dimensional models such as the generalized Nizhnik–Novikov–Veselov system, Broer–Kaup–Kupershmidt equation, dispersive long wave system, Boiti–Leon–Pempinelli model, generalized Burgers model, generalized Ablowitz–Kaup–Newell–Segur system and Maccari equation. The universal formula in Tang, Lou, and Zhang [2] can be simplified to the common formula in the present paper. Second, this method is successfully generalized to (1 + 1)-dimensional systems, such as coupled integrable dispersionless equations, shallow water wave equation, Boiti system and negative KdV model, and is able to obtain another common formula to describe suitable physical fields or potentials of these (1 + 1)-dimensional models, which is similar to the one in (2 + 1)-dimensional systems. Finally, based on the common formula for (2 + 1)-dimensional systems and by selecting appropriate multivalued functions, elastic and inelastic interactions among special dromion, special peakon, foldon and semi-foldon are investigated. Furthermore, the explicit phase shifts for all the local excitations offered by the common formula have been given, and are applied to these novel interactions in detail.

New solitary wave and periodic wave solutions to the (2 + 1)-dimensional Nizhnik–Novikov–Veselov system

In this work, the (2 + 1)-dimensional Nizhnik–Novikov–Veselov (NNV) system is studied. The tanh method, the extended tanh method, and the cosh–sinh method are used to investigate this system. By means of these methods, new solitary wave and periodic wave solutions are determined.

Novel types of interactions between solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system

Using the extended tanh-function method (ETM) based on mapping method, the variable separation solutions of the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system are derived. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the variable separation solution and by selecting appropriate functions, new types of interactions among semi-foldons and special dromions, peakons, foldons constructed by multi-valued functions are investigated. Moreover, The explicit phase shifts for all the local excitations offered by the quantity U have been given, and are applied to these novel interactions in detail.

Fractal and chaotic patterns of Nizhnik–Novikov–Veselov system derived from a periodic wave solution

Starting from an extended mapping method and a linear variable separation method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for ( 2 + 1 )-dimensional Nizhnik–Novikov–Veselov system (NNV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this Letter, we find that some novel and interesting localized coherent excitations such as stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns and chaotic patterns also exist in the NNV system as considering appropriate boundary conditions and/or initial qualifications.

Variable separation solution and interaction properties for the [formula omitted]-dimensional modified Nizhnik–Novikov–Veselov system

By means of a special variable separation approach, three interesting new variable separation solutions with certain arbitrary functions for modified Nizhnik–Novikov–Veselov system have been obtained. Plenty of localized excitations not only single-valued but also multivalued, such as the solitoffs, dromions, multidromions, lumps, breathers, instantons, multivalued solitary waves, etc., have been constructed on the basis of selecting the functions appropriately. Localized solutions of the MNNV system both with and without completely elastic interaction may be generated by taking arbitrary functions as multisoliton or multikink solutions of appropriate ( 1 + 1 ) -dimensional integrable equations.

Variable separation solutions obtained from Darboux Transformations for the asymmetric Nizhnik–Novikov–Veselov system

The use of a seed solution with some arbitrary functions for the asymmetric Nizhnik–Novikov–Veselov system in the first step Darboux transformation yields the variable separable solutions with two space-variable separated functions. The more variable separated functions which are not arbitrary can be introduced by using the Darboux transformation repeatedly. The Nth step Darboux transformation (for arbitrary N) with arbitrary number of space–variable separated functions is explicitly written down by means of the Pfaffian. The ”universal” variable separation formula which is valid for a diversity of (2 + 1)-dimensional integrable systems can be obtained from a particular reduction of the solutions constructed from the second step Darboux transformation. A new saddle-type ring soliton solution with completely elastic interaction and nonzero phase shifts is also studied in this paper.

Multisoliton Solutions of the (3+1)-Dimensional Nizhnik–Novikov–Veselov Equation**The project supported by Outstanding Youth Foundation, Research Fund for the Doctoral Program of Higher Education of China and National Natural Science Foundation of China

Using the standard truncated Painlevé analysis, we can obtain a Bäcklund transformation of the (3+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equation and get some (3+1)-dimensional single-, two- and three-soliton solutions and some new types of multisoliton solutions of the (3+1)-dimensional NNV system from the Bäcklund transformation and the trivial vacuum solution.