Dependent Variable Transformation Research Articles

Painlevé analysis and exact solutions of the resonant Davey–Stewartson system

It is shown that the resonant Davey–Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, “Y” soliton solution, “V” soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

THE PAINLEVÉ INTEGRABILITY AND N-SOLITONIC SOLUTION IN TERMS OF THE WRONSKIAN DETERMINANT FOR A VARIABLE-COEFFICIENT VARIANT BOUSSINESQ MODEL OF NONLINEAR WAVES

A variable-coefficient variant Boussinesq (VCVB) model describes the propagation of long waves in shallow water, the nonlinear lattice waves, the ion sound waves in plasmas, and the vibrations in a nonlinear string. With the help of symbolic computation, a VCVB model is investigated for its integrability through the Painlevé analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the dependent variable transformations are obtained through which the VCVB model is bilinearized. Furthermore, the corresponding N-solitonic solutions with graphic analysis are given by the Hirota method and Wronskian technique. Additionally, a bilinear Bäcklund transformation is constructed for the VCVB model, by which a sample one-solitonic solution is presented.

Breather Lattice Solutions to Negative mKdV Equation

In this paper, dependent and independent variable transformations are introduced to solve the negative mKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the negative mKdV equation, including breather lattice solution and periodic wave solution.

Two-loop QCD helicity amplitudes for ([formula omitted])-jet production in deep inelastic scattering

We derive the two-loop QCD helicity amplitudes for the processes ℓq→ℓqg (ℓq¯→ℓq¯g) and ℓg→ℓqq¯, which are the partonic reactions yielding (2+1)-jet final states in deep inelastic lepton nucleon scattering. The amplitudes are obtained by analytic continuation of the known helicity amplitudes for e+e−→qq¯g. We separate the infrared divergent and finite parts of the amplitudes using Catani's infrared factorization formula. The analytic results for the finite parts of the amplitudes are expressed in terms of one- and two-dimensional harmonic polylogarithms. To evaluate these functions numerically, we list in detail the non-trivial (and kinematic region dependent) variable transformations one needs to perform.

A systematic method to construct Hirota’s transformations of continuous soliton equations and its applications

Hirota’s bilinear method is a powerful tool for obtaining a wide class of exact solutions of soliton equations. The crucial step of this method is to find a suitable dependent variable transformation, i.e. Hirota’s transformation, which transforms a soliton equation into a Hirota bilinear equation. In this paper, a systematic method to construct Hirota’s transformations of continuous soliton equations is proposed. And some examples are given to illuminate the availability of this method. In addition, a new two-soliton solution of a coupled nonlinear Schrödinger equation is obtained.

Exact Solutions to Short Pulse Equation

In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.

Exact Solutions to Degasperis–Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.

N-Soliton Solutions of General Nonlinear Schrödinger Equation with Derivative

The bilinear equation of the general nonlinear Schrödinger equation with derivative (GDNLSE) and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrödinger equation with derivative (DNLSE) and its multisoliton solutions are given by reduction.

Exact solutions to the sine-Gordon-type equations

In this paper, the sine-Gordon-type equations, including single sine-Gordon equation, double sine-Gordon equation and triple sine-Gordon equation are studied by introducing certain dependent variable transformations and the projective Riccati equation method. Several new exact solutions are obtained.

Comment on “Loop Soliton Solutions of String Interacting with External Field”

where q, r, and s are physical observables, x and t representing spaceand time-like independent variables. Taking q 1⁄4 Z, r 1⁄4 X þ iY , and s 1⁄4 X iY with i 1⁄4 1, eq. (4) is transformed to eq. (1) provided 1⁄4 xþ t and 1⁄4 x t. Singularity structure analysis of eq. (4) has recently been investigated. The associated Backlund transformation has been constructed and Hirota’s bilinearization also given through dependent variable transformations. As a result, q, r, and s are expressed as follows r 1⁄4 G=F; s 1⁄4 H=F; q 1⁄4 2@tðlnFÞ: ð5Þ We assume r 1⁄4 s, ð?Þ denoting complex conjugation, and we transform eq. (5) to a new one by looking for some soliton solutions with the following asymptotic behavior jrj ! 0; q ! x; as jxj ! 1: ð6Þ We may then write

On Some Classes of Breather Lattice Solutions to the sinh-Gordon Equation

In this paper, dependent and independent variable transformations are introduced to solve the sinh- Gordon equation by using the knowledge of the elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the sinh-Gordon equation, including breather lattice solutions and periodic wave solutions.

Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation

In this paper, by virtue of symbolic computation, the investigation is made on a generalized variable-coefficient higher-order nonlinear Schrödinger equation with varying higher-order effects and gain or loss, which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide. A modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable-coefficient bilinear form. Based on the formal parameter expansion technique, the multi-soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints. A Bäcklund transformation in bilinear form is also obtained for the first time in this paper. Finally, discussions on the analytic soliton solutions are given and various propagation situations are illustrated.

Breather solutions and breather lattice solutions to the sine-Gordon equation

In this paper, dependent and independent variable transformations are introduced to solve the sine–Gordon (SG) equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained for the (SG) equation, including breather solutions and breather lattice solutions.

Exact solutions for the mixed AKNS equation: Hirota approach

The mixed AKNS equation, which contains an isospectral term and a nonisospectral term, is an important nonlinear evolution equation in the AKNS system. So searching for its exact solutions is vital both for the AKNS system and in mathematical sense. In this paper, the corresponding Lax pair was given, the bilinear forms of the mixed AKNS equation were obtained through introducing the transformation of dependent variables. By using Hirota’s bilinear method, the N-soliton solutions were obtained.

Edible biomass production from some important browse species in the Sahelian zone of West Africa

Acacia senegal, Guiera senegalensis and Pterocarpus lucens, browse species important in the Sahelian zone of Burkina Faso were studied by the estimation of their phenological variation over time and the evaluation of edible biomass production, total and accessible directly to animals. Biomass production was also estimated using dendrometric parameters. All the three species started the foliation phase as soon as the rains started. A. senegal and P. lucens flowered before G. senegalensis and A. senegal lost leaves earlier. The fruiting phase lasted 6–7 months for all species. Accessible edible biomass varied according to the animal species, the plant species and the height of plants. G. senegalensis showed the highest proportion of accessible biomass, but P. lucens had higher total edible biomass. Goats browsing at higher height had more edible biomass at their disposal. The accessible edible biomass was weakly correlated with tree parameters, while crown diameter was the best parameter to predict total edible biomass production, with R 2 varying from 90% ( G. senegalensis) to 98% ( P. lucens) in log 10 transformation of dependent and independent variables. The single species models developed could be applied in similar agro-ecological zones, taking into account the height stratification of plants. Further investigations on others species are needed to be able to estimate total biomass available for browsing.

A systematical way to find breather lattice solutions to the positive mKdV equation

In this paper, dependent and independent variable transformations are introduced to solve the positive mKdV equation systematically by using knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the positive mKdV equation, including many kinds of breather lattice solutions.

Envelope breather solution and envelope breather lattice solutions to the NLS equation

In this Letter, dependent and independent variable transformations are introduced to solve the nonlinear Schrödinger (NLS) equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the NLS equation, including many kinds of envelope breather solutions and envelope breather lattice solutions.

Double Wronskian Soliton Solution for Mixed AKNS System

The Lax pair of the mixed Ablowitz–Kap–Newell–Segur (AKNS) system is obtained from compatibilitycondition. Hirota's bilinear form is derived by some dependent variabletransformation. Moreover, by means of the Wronskian technique, thedouble Wronskian form of soliton solutions are found. Specially, thetwo-soliton solution is presented.

Homoclinic orbits of the Davey-Stewartson equations

The homoclinic solutions problem of the Davey-Stewartson (DS) Equations were studied. By using the Hirota’s bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the dependent variable transformation. The homoclinic orbits of the Davey-Stewartson Equations were discussed.