Multi-linear Variable Separation Research Articles

New interaction solutions of the (2+1)-dimensional NNV-type system and fusion phenomena

Abstract By means of the multilinear variable separation(MLVS) approach, new interaction solutions with low-dimensional arbitrary functions of the (2+1)-dimensional NNV-type system are constructed. Four-dromion structure, ring-parabolic soliton structure and corresponding fusion phenomena for the physical quantity U=ln f_{xy} are revealed for the first time. This MLVS approach can also be used to deal with the (2+1)-dimensional Sasa-Satsuma system.

Fusion and fission phenomena in a (2+1)-dimensional Sawada-Kotera type system

In this study, we extend the generalized multilinear variable separation approach to a fifth-order nonlinear evolution equation. By performing asymptotic analysis on the variable separation solution, which is composed of three lower-dimensional functions, we identify a resonant regime governing dromion-dromion/solitoff interactions. In the case of two-dromion interactions, elastic, inelastic, and completely inelastic collisions are possible, while for the dromion-solitoff interaction only inelastic and completely inelastic collisions are permitted. Furthermore, we derive two types of semi-rational solutions from the quadratic function ansatz. In particular, in the scenario of a completely resonant collision between a lump and a line-soliton pair, the lump separates from one line soliton and exists briefly before merging with the other soliton, forming a localized lump in both time and space dimensions. The fusion or fission phenomena between the dromion-dromion/solitoff interaction and the lump-line soliton interaction are shown graphically.

New Multilinear Variable Separation Solutions of the (3 + 1)-Dimensional Burgers Hierarchy

The multilinear variable separation (MLVS) approach has been proven to be very useful in solving (2 + 1)-dimensional integrable systems. Taking the (3 + 1)-dimensional Burgers hierarchy as an example, we extend the MLVS approach to a whole family of (3 + 1)-dimensional Burgers hierarchy. New exact solutions and universal formulas are obtained, which lead to abundant (3 + 1)-dimensional coherent structures. In particular, two ring-type soliton molecules and their interactions are shown in detail. We also generalize the MLVS results of the (3 + 1)-dimensional Jimbo–Miwa (JM) equation and modified JM equation.

Non-compatible fully [formula omitted] symmetric Davey–Stewartson system: Localized excitation and folded solitary wave

This paper studies a non-compatible fully PT symmetric Davey–Stewartson system which models the evolution of optical wave packet in nonlinear optics. “Non-compatible” means the first two equations of the system are inconsistent that do not need to comply with extra constraint like the compatible case. We employ the multi-linear variable separation approach to derive variable separation solution for this system. The obtained solutions contain four separated free functions, render enough freedom for us to generate diverse solitons. Firstly, we construct two kinds of localized excitations: lump-lattice and periodic-lattice structure. Secondly, we obtain four typical folded solitary waves, i.e. worm-dromion shaped, fin shaped and octopus shaped waves, by introducing suitable multi-valued functions. Lastly, we systematically analyze the interaction between two and three foldons, and give the classification of it. To our knowledge, this is the first time variable separation solution been reported in PT symmetric system. The basic idea of separating variables that used in this paper can be extended to other symmetric systems.

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

New variable separation solutions and localized waves for (2+1)-dimensional nonlinear systems by a full variable separation approach

A full variable separation approach is firstly proposed for (2+1)-dimensional nonlinear systems by extending the well-established multilinear variable separation approach through the assumption that the expansion function is composed of full variable separated functions, namely, functions with respect to only one spacial or temporal argument. Taking the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, the Nizhnik-Novikov-Veselov equation and the dispersive long wave equation as examples, new full variable separation solutions are obtained with several arbitrary one dimensional functions. Especially, a common formula for some suitable physical quantities is discovered. By taking the arbitrary functions in different explicit expressions, the solutions can be used to describe plentiful novel nonlinear localized waves, which might be non-travelling waves as the spacial and temporal variables are fully separated into different functions. In particular, some new hybrid solitary waves, which can pulsate periodically, appear and/or decay with an adjustable lifetime, are discovered through the on-site interactions between a doubly periodic wave and a ring soliton, a four-humped dromion and a four-humped lump, and a doubly periodic wave and a cross type solitary wave. Nonlinear wave structures and their dynamical behaviours are discussed and graphically displayed in detail.

Bosonization, symmetry reductions, mapping and deformation method for B-extension of Sawada–Kotera equation

An extended (2+1)-dimensional shallow water wave (SWW) model governs the evolution of nonlinear shallow water wave propagation in two spatial and a temporal coordinate. The multi-linear variable separation approach is applied to the SWW equation. The variable separation solution consisting of two arbitrary functions is given. By choosing the arbitrary functions as the exponential and trigonometric forms, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions and periodic waves are graphically and analytically studied. The results enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Novel soliton molecules and interaction wave solutions in a (2+1)-dimensional Sawada–Kotera equation: a multi-linear variable separation method

Novel soliton molecules and interaction wave solutions in a (2+1)-dimensional Sawada–Kotera equation: a multi-linear variable separation method

Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear system

In this paper, we propose a new variable separation method that does not need the Hirota’s bilinear form and directly gives the analytic form of the solution u instead of its potential uy. This new method covers the N-soliton solution obtained by the Hirota’s direct method and the multi-valued solution of the multi-linear variable separation approach(MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the “direct separation approach” (DSA). Taking the extended (3+1)-dimensional Kadomtsev–Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.

Solitons solutions to the high-order dispersive cubic–quintic Schrödinger equation in optical fibers

In this paper, solitons solutions of higher-order dispersive cubic–quintic Schrödinger equationincluding third-order as well as fourth-order derivatives with respect to time, that describes the dynamics of ultrashort pulses in optical fibers are investigated in detail. In this respect,a solution procedure in the locality of applied mathematics called the hyperbolic function method is appliedusing multi-linear variable separation approach (MLVSA). As an outcome, a bunch of soliton solutions isderived in conjunction with plotting dark and periodic wave solutions. The credibility of the results is examined by setting each solution back into its governing equation. Through portraits, different forms of wave solutions are depicted. Moreover, the restrictions on the parameters are also given for the existence of the obtained solutions.

Novel soliton molecules and wave interactions for a (3 + 1)-dimensional nonlinear evolution equation

New wave excitations are revealed for a (3 + 1)-dimensional nonlinear evolution equation to enrich nonlinear wave patterns in nonlinear systems. Based on a new variable separation solution with two arbitrary variable separated functions obtained by means of the multilinear variable separation approach, localized excitations of N dromions, $$N\times M$$ lump lattice and ring soliton lattice are constructed. In addition, it is observed that soliton molecules can be composed of diverse “atoms” such as the dromions, lumps and ring solitons, respectively. Elastic interactions between solitons and soliton molecules are graphically demonstrated.

New variable separation solutions and wave interactions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

New variable separation solutions with arbitrary low dimensional functions are obtained for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation by means of the multilinear variable separation approach. A general cluster of N×M dromions is explicitly presented, from which novel interesting interactions between dromions, especially dromion molecules are discussed.

Interacting waves of Davey–Stewartson III system

Modulational unstable regions of the Davey–Stewartson (DS) III system have been determined from the generalized dispersion relation associating the frequency and wavenumber of the modulating perturbations. By means of the multilinear variable separation approach, the variable separation solution for the DS III equation is obtained with two arbitrary functions of (x,t) and two arbitrary functions of (y,t), which can be utilized to generate various (2+1)-dimensional localized excitations. Particular attention is paid on the interacting waves between periodic multivalued foldons and single-valued dromions, which can be viewed as periodic extensions of single foldon–dromion excitations.

Optical soliton wave solutions to the resonant Davey–Stewartson system

We investigate the resonant Davey–Stewartson (DS) system. The resonant DS system is a natural \((2+1)\)-dimensional version of the resonant nonlinear Schrodinger equation. Traveling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, improved \(\tan (\phi /2)\)-expansion method (ITEM) and generalized (G′/G)-expansion method for seeking more exact solutions via the resonant Davey–Stewartson system. These methods are direct, concise and simple to implement compared to other existing methods. The exact particular solutions containing four types of solutions, i.e., hyperbolic function, trigonometric function, exponential and solutions. We obtained further solutions comparing these methods with other methods. The results demonstrate that the aforementioned methods are more efficient than the multi-linear variable separation method applied by Tang et al. (Chaos Solitons Fractals 42:2707–2712, 2009). Recently the ITEM was developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving the nonlinear problems.

Folded solitary waves of the Boiti–Leon–Pempinelli system

The folded solitary waves are the multivalued localized excitations. Traditionally, to describe these complicated structures, one needs to find the “universal” formula through the multilinear variable separation approach. In this paper, the Boiti–Leon–Pempinelli system is first reduced to the unsteady convection–diffusion equation according to the homogeneous balance principle. And then, the Lie group theory is applied to this equation and the similarity reduction equations, and the similarity solutions are obtained. Thus, the abundant folded solitary waves of the Boiti–Leon–Pempinelli system are investigated. These results illustrate that the multivalued structures can be derived through another skill.

Be careful with variable separation solutions via the extended tanh-function method and periodic wave structures

We analyze the extended tanh-function method to realize variable separation, however, we find that various different solutions obtained by this method are seriously equivalent to the general solution derived by the multilinear variable separation approach. In order to illustrate this point, we take a general (2 + 1)-dimensional Korteweg–de Vries system in water for example. Eight kind of variable separation solutions for a general (2 + 1)-dimensional Korteweg–de Vries system are derived by means of the extended tanh-function method and the improved tanh-function method. By detailed investigation, we find that these seemly independent variable separation solutions actually depend on each other. It is verified that many of so-called new solutions are equivalent to one another. Based on the uniform variable separation solution, abundant localized coherent structures can be constructed. However, we must pay our attention to the solution expression of all components to avoid the appearance of some un-physical related and divergent structures: seemly abundant structures for a special component are obtained while the divergence of the corresponding other component for the same equation appears.

Painlevé Integrability and a New Exact Solution of the Multi-Component Sasa-Satsuma Equation

Abstract In this article, Painlevé integrability of the multi-component Sasa-Satsuma equation is confirmed by using the standard WTC approach and the Kruskal simplification. Then, by means of the multi-linear variable separation approach, a new exact solution with lower-dimensional arbitrary functions is constructed. For the physical quantity U = ∑ i = 1 N ∑ j = i N a i j p i p j = − 3 2 β F x G y ( F + G ) 2 , $U\; = \;\sum\nolimits_{i\; = \;1}^N \sum\nolimits_{j\; = \;i}^N {a_{ij}}{p_i}{p_j}\; = \; - \;\frac{3}{{2\beta }}\frac{{{F_x}{G_y}}}{{{{(F\; + \;G)}^2}}},$ new coherent structure which possesses peakons at x-axis and compactons at y-axis is illustrated both analytically and graphically.

Painlevé Property, Bäcklund Transformations and Rouge Wave Solutions of (3 + 1)-Dimensional Burgers Equation

Burgers equation is the simplest one in soliton theory, which has been widely applied in almost all the physical branches. In this paper, we discuss the Painlevé property of the (3+1)-dimensional Burgers equation, and then Bäcklund transformation is derived according to the truncated expansion of the obtained Painlevé analysis. Using the Bäcklund transformation, we find the rouge wave solutions to the equation via the multilinear variable separation approach. And we also give an exact solution obtained by general variable separation approach, which is proved to possess abundant structures.

Variable separation solutions and interacting waves of a coupled system of the modified KdV and potential BLMP equations

Abstract The multilinear variable separation approach (MLVSA) is applied to a coupled modified Korteweg–de Vries and potential Boiti–Leon–Manna–Pempinelli equations, as a result, the potential fields u y and v y are exactly the universal quantity applicable to all multilinear variable separable systems. The generalized MLVSA is also applied, and it is found that u y ( v y ) is rightly the subtraction (addition) of two universal quantities with different parameters. Then interactions between periodic waves are discussed, for instance, the elastic interaction between two semi-periodic waves and non-elastic interaction between two periodic instantons. An attractive phenomenon is observed that a dromion moves along a semi-periodic wave.

Dynamics of dispersive long waves in fluids

We obtain many different variable separation solutions for (2+1)-dimensional variable coefficient dispersive long wave equation by means of five different methods, including the multilinear variable separation approach, the projective Ricatti equation method, the extended projective Ricatti equation method, the extended tanh-function method and the improved tanh-function method. However, by careful analysis, we find that variable separation solution obtained by the multilinear variable separation approach includes all variable separation solutions obtained by other four direct methods. Thus variable separation solution for (2+1)-dimensional variable coefficient dispersive long wave equation exists a uniform form. Based on this uniform variable separation solution, we discuss the completely elastic interaction between foldons, the non-completely elastic interaction between bell-like semi-foldon, peaked semi-foldon and foldon, and the completely non-elastic interaction between bell-like semi-foldon and peaked semi-foldon. These results are helpful to analyze more precisely nonlinear and dispersive long gravity waves traveling in two horizontal directions, such as the bubbles on (or under) a fluid surface and folded waves in various ocean waves.