Nonlinear Superposition Formula Research Articles

Bell polynomials and superposition wave solutions of Hirota–Satsuma coupled KdV equations

In this article, the bilinear form, the bilinear Bäcklund transformation, Lax pair, the integrability, infinite conservation laws, nonlinear superposition formula and superposition wave solutions of Hirota–Satsuma coupled KdV (HSCKdV) equations are studied, which can help us get more properties, increase the diversity of solutions and get more new phenomena. The bilinear form, the bilinear Bäcklund transformation and Lax pair of HSCKdV equations are constructed by using the Bell polynomials approach. Based on this, HSCKdV system is integrable in the sense of Lax pair. According to the obtained bilinear Bäcklund transformation, infinite conservation laws and nonlinear superposition formula are derived. And some superposition wave solutions of HSCKdV equations can be attained by the nonlinear superposition formula. With the help of the symbolic calculation system Mathematica, the properties of the superposition wave solutions are analyzed. What is more, in addition to many properties obtained, the relationship between the solutions is found via the derived nonlinear superposition formula, and mixed solutions containing different functions and arbitrary functions are constructed. By using the Bell polynomial method and studying the integrability and other properties of HSCKdV equations, as well as their exact solutions, it can be found that solitons maintain their shape unchanged during motion and are not destroyed. Such waves have very important applications as a communication method. Furthermore, it has been observed that the results attained in this work have not been presented before.

Bäcklund transformation and nonlinear superposition formula for the two-component short pulse equation

By means of the reciprocal transformation and the associated equation, a Bäcklund transformation (BT) containing both independent and dependent variables is worked out and studied for the two-component short pulse (2SP) equation. Based on it, the corresponding nonlinear superposition formula (NSF) is also constructed. Through reducing the BT and NSF of the 2SP equation, the BTs and NSFs for the SP equation, the focusing complex short pulse (cSP) equation and the defocusing cSP equation are obtained respectively. By analyzing the range of parameters in the solution, various solutions including soliton solitons, cuspon solitons, loop solitons, breather solutions are divided and their interaction are considered.

A note on Bäcklund transformations for the Harry Dym equation

The Bäcklund transformations are revisited for the Harry Dym equation. The associated nonlinear superposition formula, which consists of two equations — one for the field variable and the other for the spatial variable, is also presented. Interpreted as discrete systems, nonlinear superposition formula is identified as the lattice potential modified Korteweg–de Vries equation and a special case of Q1 equation in the ABS list. A particular limit of the nonlinear superposition formula is considered and the resulting Bäcklund transformation is shown to be related to the finite transformation of the nonlocal symmetries.

On a fermionic extension of [formula omitted] equation

A fermionic extension of K(−1,−2) equation, alias a super K(−1,−2) equation, proposed by Tempesta et al. (2003), is converted to the N=1 supersymmetric Korteweg–de Vries equation via invertible transformations involving both independent and dependent variables. As implementation of this intimate connection, some integrable properties are established for the fermionic extension of K(−1,−2) equation, including a linear spectral problem in terms of 3 × 3 matrices, a Bäcklund transformation, a nonlinear superposition formula, infinitely many conservation laws, as well as its bi-Hamiltonian formulation.

A super KdV equation of Kupershmidt: Bäcklund transformation, Lax pair and related discrete system

A super extension of the Korteweg-de Vries (KdV) equation proposed by Kupershmidt is studied. Its Bäcklund transformation is explicitly presented and a Lax pair is derived, so the integrability of this system is confirmed. The corresponding nonlinear superposition formula is constructed from the Bäcklund transformation and is interpreted as a fully discrete system, whose integrability is demonstrated by verifying its consistency around the cube (CAC) property and supplying a zero-curvature representation.

M -Breather, Lumps, and Soliton Molecules for the 2 + 1 -Dimensional Elliptic Toda Equation

The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

M-breather, M-lump, breather molecules and their interaction solutions for a (2+1)-dimensional KdV equation

Date-Jimbo-Kashiwara-Miwa (DJKM) equation is an integrable (2+1)- dimensional extension of the KdV equation and describes two-dimensional nonlinear dispersive waves. In this paper, we first derive the M-breather solution in terms of determinants for the DJKM equation applying the nonlinear superposition formula and analyze the dynamical properties of the breathers. The M-lump waves for the DJKM equation are obtained through the full degeneration of the breathers and hybrid solutions composed of line solitons, breathers and lumps are constructed. By using the velocity resonance mechanism, we also show that the DJKM equation possesses some resonant structures for breathers and solitons such as breather molecules and breather-soliton molecules. Furthermore, based on the N-soliton solution, the interactions between breather/breather-soliton molecules and breathers/lumps are investigated.

A Note on a Three-Dimensional Bäcklund Transformation

The Backlund transformation (BT) for a three-dimensional nonlinear wave equation and its nonlinear superposition formula are studied in this note. We prove that the three dimensional Backlund transformation obtained by Leibbrandt, et al. can be decomposed into three two-dimensional BTs. Some results on the N-dimensional Liouville equation are also discussed in the article.

The short pulse equation: Bäcklund transformations and applications

AbstractA Bäcklund transformation (BT), which involves both independent and dependent variables, is established and studied for the short pulse (SP) equation. Based it, the nonlinear superposition formulae for 2‐, 3‐, and 4‐BT are presented. The general result for the composition of ‐BTs is achieved and given in terms of determinants. As applications, various solutions including loop solitons, breather solutions, and their interaction solutions are worked out.

The modified Camassa–Holm equation: Bäcklund transformation and nonlinear superposition formula**Dedicated to the memory of Jon Nimmo.

This paper is concerned with the modified Camassa–Holm equation and its Bäcklund transformation. With the aid of reciprocal transformation and the associated modified Camassa–Holm equation, a Bäcklund transformation, which involves both dependent and independent variables, is constructed for the modified Camassa–Holm equation. The related nonlinear superposition formula is worked out and applications of both Bäcklund transformation and nonlinear superposition formula are considered.

An analytic approach to constructing Bäcklund transformations and exact solutions to nonlinear wave equations in non-polynomial form

Bäcklund transformation (BT) is an intrinsic property of nonlinear integrable system. Generally, it is an interesting and challenging work to investigate BT of a nonlinear system, especially for the non-polynomial form. In this paper, we introduce an analytic method for constructing BTs of the generalized nonlinear wave equations (NLWEs) of the form utt=auxx+h(u). The BTs with arbitrary parameters are provided explicitly, so the integrability of the equation is verified accordingly. Then, the nonlinear superposition formulas (NSFs) of the NLWEs are given in terms of such BTs, and the infinite number of exact explicit solutions to the equations are obtained based on the BTs and the NSFs. Furthermore, the BTs of the other types of NLWEs of the form uxt=h(u) can be provided directly by the variable transformation method.

Lax Pair, Improved Γ $\varGamma $ -Riccati Bäcklund Transformation and Soliton-Like Solutions to Variable-Coefficient Higher-Order Nonlinear Schrödinger Equation in Optical Fibers

In this paper, the higher-order nonlinear Schrodinger equation with variable-coefficient for the propagation of femtosecond pulses in optical fibers is analytically investigated. Lax pair is constructed via the Ablowitz-Kaup-Newell-Segur system, an infinite number of conservation laws are derived based on the Lax pair. Introducing an auxiliary function, an improved $\varGamma $ -Riccati Backlund transformation is presented, which can generate successively nonlinear superposition formula. Moreover, one and two soliton-like solutions are obtained.

Lax pair and lump solutions for the (2+1)-dimensional DJKM equation associated with bilinear Bäcklund transformations

We aim to explore exact solutions and integrable properties to the (2+1)-dimensional DJKM equation. Based on the bilinear Backlund transformation, we first furnish Lax pair and complex exponential wave function solutions, and then give complexitons or hyperbolic function solutions. Moreover, via the nonlinear superposition formula, the construction procedure for presenting rational solutions is improved. The key step is that all the involved parameters are extended to the complex field. In particular, we show that the (2+1)-dimensional DJKM equation possesses a general class of lump solutions when $${\sigma }^2=-1$$.

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the supersymmetric Sawada-Kotera (SSK) equation. The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.

Bäcklund–Darboux transformations and discretizations of N = 2 a = −2 supersymmetric KdV equation

The N=2a=−2 supersymmetric KdV equation is studied. A Darboux transformation and the corresponding Bäcklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation. The Bäcklund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems. The continuum limits of these discrete systems are also considered.

Solitonic properties for a forced generalized variable-coefficient Korteweg-de Vries equation for the atmospheric blocking phenomenon

In this paper, investigation is given to a forced generalized variable-coefficient Korteweg-de Vries equation for the atmospheric blocking phenomenon. Based on the Lax pair, under certain variable-coefficient-dependent constraints, we present an infinite sequence of the conservation laws. Through the Riccati equations obtained from the Lax pair, a Wahlquist-Estabrook-type Bäcklund transformation (BT) is derived, based on which the nonlinear superposition formula as well as one- and two-soliton-like solutions are obtained. Via the truncated Painlevé expansion, we give a Painlevé BT, along with the one-soliton-like solutions. With the Painlevé BT, bilinear forms are constructed, and we get a bilinear BT as well as the corresponding one-soliton-like solutions. Bell-type bright and dark soliton-like waves and kink-type soliton-like waves are observed, respectively. Graphic analysis shows that (1) the velocities of the soliton-like waves are related to h(t), d(t), f(t) and R(t), while the soliton-like wave amplitudes just depend on f(t), and (2) with the nonzero f(t) and R(t), soliton-like waves propagate on the varying backgrounds, where h(t), d(t) and f(t) are the dispersive, dissipative and line-damping coefficients, respectively, R(t) is the external-force term, and t is the scaled time coordinate.

Bäcklund transformations of Zn-sine-Gordon systems

In this paper, from the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon systems which contain many multi-component sine-Gordon type and sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon equations which can generate new solutions from seed solutions. To see the [Formula: see text]-systems clearly, we consider the [Formula: see text]-sine-Gordon and [Formula: see text]-sine-Gordon equations explicitly including their Bäcklund transformations, the nonlinear superposition formula and Lax pairs.

A Darboux Transformation for Ito Equation

Abstract A system proposed by Ito is reconsidered. The corresponding Darboux transformation is presented explicitly. The resulted Bäcklund transformation is shown to be equivalent to the one found by Hirota. Also, a nonlinear superposition formula, which is of differential-algebraic, is obtained.

The New Infinite Sequence Solutions of Multiple Sine-Gordon Equations

By the function transformation and the first integral of the ordinary differential equations, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is researched, and the new solutions are obtained. First, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is changed to the problem of solving the solutions of the nonlinear ordinary differential equation. Second, with the help of the B?cklund transformation and the nonlinear superposition formula of solutions of the first kind of elliptic equation and the Riccati equation, the new infinite sequence soliton-like solutions of two kinds of sine-Gordon equations are constructed.

Darboux Transformation for Drinfel'd–Sokolov–Wilson Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11271366, 11331008 and 11401572, and the Fundamental Research Funds for Central Universities

A coupled system known as the Drinfel'd–Sokolov–Wilson equation is reexamined. With the help of a Lax operator of fourth order, its proper Darboux transformation is constructed. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation and some solutions are calculated.