Variable Separation Solutions Research Articles

New coherent structures and interaction behavior for the new [formula omitted]-dimensional Korteweg-de Vries equation

A new type of variable separation solutions for the (2+1)-dimensional Korteweg–de Vries equation is derived by means of an improved mapping approach. The lump solutions are constructed and the interaction phenomena between a lump wave and muliti-kink waves are discussed, and some of these solutions exhibit a rich dynamics, with a wide variety of qualitative behavior and structures that are exponentially localized, showing some novel features and interesting behaviors.

A novel approach to model nance drying considering its shrinkage

Air drying is a viable preservation method to boost the accessibility of nance, a seasonal drupe with very short shelf life; however, information about nance drying is scarce, especially from a mass transfer (MT) standpoint. Besides, the lack of simple models considering the stone fraction (radius of inner stone/radius of nance) and the mesocarp shrinkage in drupes has caused most investigations rely on sphere and cube geometries, contributing to the unreliable estimation of MT properties. This study developed a discrete state-space (SS) formulation with analytical solution to describe nance drying which, unlike a variable separation (VS) solution, can consider a variable stone fraction, caused by outer boundary moving while flesh-pit interface remains still. Nance fruits were air dried (60, 70 and 80 °C) to obtain their moisture kinetics and shrinking behavior. Drying data were used to estimate water diffusivity (D) under rigid (VS and SS solutions) and shrinkable (SS and numerical solution) mesocarp assumptions. Nances lost about 60% of their initial volume at the end of drying, while stone fraction increased from 0.44 to 0.62. D values ranged from 5.3 to 6.7 (×10–10) m2/s with SS approach and variable stone fraction. The major source of error in D was not the formulation (SS or VS) when a rigid mesocarp was considered, but the values assigned to the nance radius (length for diffusion) and stone fraction, an initial or average value along process. SS diffusivities did not differ to those estimated with the full numerical solution under variable stone fraction (p>0.05), while VS parameters were overestimated by 44–46 % as a minimum. SS formulation can be applied to hollow cylinders and other geometries with 2D or 3D MT, while able to include non-isotropic shrinkage and evolving external conditions, a problem normally reserved to pure numerical implementations, offering substantial advantages over VS equations.

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.

A novel (2+1)-dimensional Sawada-Kotera type system: multisoliton solution and variable separation solution

A novel (2+1)-dimensional Sawada-Kotera type system: multisoliton solution and variable separation solution

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.

Periodic, n-soliton and variable separation solutions for an extended (3+1)-dimensional KP-Boussinesq equation

An extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation is studied in this paper to construct periodic solution, n-soliton solution and folded localized excitation. Firstly, with the help of the Hirota’s bilinear method and ansatz, some periodic solutions have been derived. Secondly, taking Burgers equation as an auxiliary function, we have obtained n-soliton solution and n-shock wave. Lastly, we present a new variable separation method for (3+1)-dimensional and higher dimensional models, and use it to derive localized excitation solutions. To be specific, we have constructed various novel structures and discussed the interaction dynamics of folded solitary waves. Compared with the other methods, the variable separation solutions obtained in this paper not only directly give the analytical form of the solution u instead of its potential u_y, but also provide us a straightforward approach to construct localized excitation for higher order dimensional nonlinear partial differential equation.

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.

Variable separation solution and multi-valued soliton of an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

Variable separation solution and multi-valued soliton of an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

Variable separation solution for an extended (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

This paper proposes a new variable separation solution for the (3+1)-dimensional nonlinear evolution equation. The new variable separation solution directly gives the analytical form of the solution u instead of its potential uy and renders a distinct way to construct localized excitation. Taking the extended (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation as an example, we test its integrability at first. Then, we analyze the elastic, inelastic head-on collision of two and three folded solitary waves by introducing several suitable multi-valued functions. Specifically, the superimposed structure of folded waves is studied, and plenty of novel patterns have been obtained.

The variable separation solution, fractal and chaos in an extended coupled (2+1)-dimensional Burgers system

The variable separation solution, fractal and chaos in an extended coupled (2+1)-dimensional Burgers system

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.

Breather, multi-shock waves and localized excitation structure solutions to the Extended BKP–Boussinesq equation

The extended BKP–Boussinesq equation is considered to construct abundant breather waves, multi-shocks waves and localized excitation solutions. We first transform the original model to its bilinear form through a logarithmic transformation relation. Then, by setting a simple ansatz as a combinations of exponential and sinusoidal functions to obtain various breather waves solutions. We successfully archive five types of breather waves and depict graphically. Taking Burger model as an auxiliary equation, we derive multi-shock waves solutions to illustrate the overtaking collisions and energy distribution of the extended model sufficiently. Finally, we keep a simple variable separable ansatz solution to derive localized excitation structures of the model. Most of these solutions are found for the first time. Furthermore, the results disclose that the new approaches are very direct, elementary, effective and can be used for many other NLPDEs, which develop the various types of dynamical properties of any wave model.

A class of lump solutions and localized excitations for the generalized (3 + 1)-dimensional KP equation

Based on symbolic computation and Hirota bilinear form, a class of lump solutions for the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation is given. As a result, the lump solution shows a new perspective to knowledge them. Meanwhile, variable separation solutions are also constructed by applying a simple ansatz method. These variable separation solutions contain the shock solution, the peakon-type breather solution and the periodic solution. They showed some special localized structures in 2D and 3D-plots.

Complex waves and their collisions of the breaking soliton model describing hydrodynamics

As one of the important physical models, the variable-coefficient breaking soliton model describes the dynamics of solitary waves and Riemann waves in hydrodynamics. Applying the two function approach into the variable-coefficient breaking soliton model, we find an exponential-form variable separation solution with two arbitrary functions, which covers many hyperbolic and trigonometric function solutions. Based on this solution and choosing suitably two arbitrary functions, complex waves including dromion pair and single dromion superposed on a line soliton background and their collisions between complex waves and between dromion and periodic wave are discussed. For the complex waves such as the dromion pair and single dromion superposed on a line soliton background and their collisions between complex waves, two components of this model possess both striking physical meanings. However, for the collision between dromion and periodic wave, one component of this model has unique physical meaning, yet the other component of the same model appears the singularity structure. Therefore, when one discusses complex waves and their collisions for one component of the model, one must take care of non-physical structures for another component of the same model lest one arbitrarily claims that he finds the so-called novel localized structures, which are actually false non-physical structures.

Construction of localized solutions to a generalized (3+1)-dimensional Burgers equation

Finding exact and analytic solutions of nonlinear system in high dimensions is a difficult but meaningful work. In this paper, by means of the symbolic package Maple, we investigate a Painlevé integrable (3+1)-dimensional generalized Burgers (gBurgers) equation. Starting from the Cole-Hopf transformation with different seed solutions, abundant localized solutions are provided, including new variable separation solutions, lumps, lump-“multiple-soliton” solutions and other interaction solutions. Specifically, the lump-two soliton solution and lump-soliton-periodic solution are depicted by the three-dimensional plots and contour plots at different times, respectively. The methods and all the resulting solutions presented in this paper can be generalized to the Painlevé integrable (N+1)-dimensional Burgers equation.

Soliton exotic collision of the (2+1)-dimensional modified dispersive water-wave system in fluid mechanics

Using the bifunction method with the exponential form solution, which includes many solutions reported in literatures, we can obtain variable separation solutions of the (2+1)-dimensional modified dispersive water-wave system in fluid mechanics. Based on the exponential-form variable separation solution, we discuss special soliton collision between two dromion-pairs constructed by one dromion and one anti-dromion for two field components of system. Moreover, we also study compacton-like soliton fusion and fission phenomena for two field components. The singularity and un-physical structure is found for one of two field components when we discuss soliton collision between two dromion-pairs for another component, thus researcher must consider the construction of localized coherent structures for all field components in order to guarantee the importance of the research for localized coherent structures based on variable separation solutions.

Diverse oscillating soliton structures for the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

A new type of variable separation solutions for the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is derived by means of an improved mapping approach. Based on the derived variable separation excitation, rich oscillating solitons such as rogue-wave, dromion, multi-dromion, solitoff, lump and fractal-type structures are presented by selecting appropriate functions of the general variable separation solution, and some of these solutions exhibit a rich dynamic, with a wide variety of qualitative behavior and structures that are exponentially localized, showing some novel features and interesting behaviors.

Interactions between exotic multi-valued solitons of the (2+1)-dimensional Korteweg-de Vries equation describing shallow water wave

Korteweg-de Vries equation governs the weakly nonlinear long wave whose phase speed reaches a simple maximum of wave with the infinite length in shallow water wave. The exponential-form variable separation solution of (2+1)-dimensional Kortweg-de Vries equation is found via the two-function method, and this solution covers many special combined solutions including sinh-cosh,sin-cos,sech-tanh,csch-coth,sec-tan and csc-cot solutions. From the exponential-form solution with choosing suitable functions, inelastic interactions between special multi-valued solitons with two loops such as anti-bell-shaped, anti-peak-shaped semifoldons and anti-foldon are graphically and analytically studied. By the asymptotic analysis, phase shift and its difference during interactions between multi-valued solitons are analytically given.