*N*-Fold Darboux transformation and solitonic interactions for a Volterra lattice system

- Xiaoyong Wen
^{1}Email author and - Xiaoyan Hu
^{2}Email author

**2014**:213

https://doi.org/10.1186/1687-1847-2014-213

© Wen and Hu; licensee Springer. 2014

**Received: **16 April 2014

**Accepted: **15 July 2014

**Published: **4 August 2014

## Abstract

Under consideration in this paper is a Volterra lattice system. Through symbolic computation, the Lax pair and conservation laws are derived, an integrable lattice hierarchy and an *N*-fold Darboux transformation (DT) are constructed for this system. Furthermore, *N*-soliton solutions in terms of determinant are generated with the resulting *N*-fold DT. Structures of the one-, two- and three-soliton solutions are shown graphically. Overtaking inelastic solitonic interactions between/among the two and three solitons are discussed by figures plotted.

## Keywords

*N*-fold Darboux transformation

*N*-soliton solutions in terms of determinantconservation lawssymbolic computation

## 1 Introduction

Explicit solutions of the nonlinear partial differential equations (NPDEs), in particular the soliton solutions, describe certain phenomena (see [1] and references therein). A soliton is a localized nonlinear wave which has particle-like properties [2]. Nonlinear differential-difference equations (NDDEs), taken as spatially discrete analogues of the NPDEs, have received certain attention [2–4]. Studies on the solitons might be divided into two categories, *i.e.*, the continuous and discrete (lattice) cases [2]. Dynamical behaviors of the solitons in the continuous and discrete cases are described by the NPDEs and NDDEs, respectively [2]. NDDEs have some applications in science [2–6]. For example, the Toda lattice [5] is the discrete approximation of the Korteveg-de Vries (KdV) equation in fluids; the discrete nonlinear Schrödinger equation [6] can describe the interaction and propagation of optical pulses in a nonlinear waveguide array; the Volterra lattice system [2, 7–13] is in connection with the spectrum of Langmuir wave in plasma dynamics.

Explicit solutions might be helpful for understanding some processes described by the NDDEs, especially the soliton solutions [2, 14]. Solitons in the discrete systems are sometimes called the lattice solitons [2]. Methods for constructing the explicit solutions of the NDDEs, such as the inverse scattering method [14–16], the Bäcklund transformation [17, 18], the Hirota method [19, 20] and the DT [21–27], have been developed. Among them, the DT is an algebraic one used to obtain the explicit solutions (especially the multi-soliton solutions) in a recursive manner [28]. The key idea of the DT method is to keep the linear eigenvalue problems of the integrable NDDEs invariant.

where ${M}_{n}=M(n,t)$ are the functions of the discrete variable *n* and time variable *t*, ${M}_{n,t}=\frac{d{M}_{n}}{dt}$. Equation (1) is in connection with the spectrum of Langmuir waves in space and laboratory plasmas [2]. References [29–32] have presented some rational, solitary-wave and periodic-wave solutions of (1). In [33], the traveling-wave solution of Volterra lattice was constructed by the optimal homotopy analysis method. Although many people have investigated Eq. (1), to our knowledge, few people have studied Eq. (1) via the *N*-fold DT. Furthermore, inelastic interaction behaviors of the discrete solitons and conservation laws for this system have not been reported previously.

Different from the previous studies, in this paper, we make further investigation on Eq. (1) via the *N*-fold DT technique [34]. By employing the AKNS (Ablowitz-Kaup-Newell-Segur) procedure [35], we construct the new Lax pair in matrix form associated with Eq. (1). Based on the derived Lax representation, we directly construct the *N*-fold Darboux matrices for Eq. (1). Outline of this paper is as follows. In Section 2, an integrable lattice hierarchy associated with Eq. (1) is given from a discrete spectral problem. In Section 3, the Lax pair and *N*-fold DT of (1) are constructed by employing the AKNS procedure. In Section 4, *N*-soliton solutions in terms of determinant are derived via the resulting *N*-fold DT, the solitonic interaction of those solutions is analyzed graphically. In Section 5, conservation laws of (1) are given. Conclusions are made in the last section.

## 2 An integrable lattice hierarchy associated with Eq. (1)

where *λ* is a spectral parameter and ${\lambda}_{t}=0$, $\beta \ne 0$ is an arbitrary constant, ${\phi}_{n}={({\phi}_{1,n},{\phi}_{2,n})}^{T}$ is a vector eigenfunction, $u={({u}_{n},{v}_{n})}^{T}$ is the potential function and *E* is the shift operator defined by $Ef(n,t)=f(n+1,t)\equiv {f}_{n+1}$, ${E}^{-1}f(n,t)=f(n-1,t)\equiv {f}_{n-1}$, $n\in Z$, $t\in R$, *T* denoting the transpose of the matrix.

When $\beta =1$, ${u}_{n}=-{v}_{n}=-{M}_{n}$, system (18) reduces to Eq. (1).

The Hamiltonian structure often guarantees the existence of infinitely many symmetries and infinitely many conserved functionals, exhibiting integrability of the equations under consideration [37]. For the obtained lattice hierarchies (12), (15) and (17), we also may construct their Hamiltonian structures. The aim of this paper is to construct *N*-fold DT and multi-soliton solutions in terms of determinant of Eq. (1). Hence, as to the detailed derivation process on how to construct Hamiltonian structures of the obtained hierarchies, we refer the reader to the work of Ma [37], here we omit them for simplification.

## 3 *N*-Fold DT of Eq. (1)

At present, more research on the Lax integrable NPDEs has been done via the *N*-fold DT [38–41], for the Lax integrable NDDEs, more research has been done by a single DT (*i.e.*, 1-fold DT) [21–27]. However, as far as we know, few studies on the NDDEs have been done by constructing the *N*-fold DT. Although the *N*-fold DT can be interpreted as a superposition of the 1-fold DT, comparing with the 1-fold DT, the biggest advantage of *N*-fold DT is that we can obtain the relationships between the new multi-soliton solutions and the seed solutions without complicated iterations, so it is meaningful to generalize the *N*-fold DT technique from NPDEs to NDDEs.

*i.e.*,

*N*-fold DT of (1). Hereby, we construct a special ${T}_{n}$ as follows:

*n*and

*t*. ${a}_{n}^{(j)}$, ${b}_{n}^{(j)}$ can be determined by the following linear algebraic system:

and ${\phi}_{n}=({\phi}_{1,n},{\phi}_{2,n})$ is a solution of (20) and (21). When the $2N+1$ parameters ${\lambda}_{i}$ (${\lambda}_{i}\ne {\lambda}_{j}$, $i\ne j$) are suitably chosen so that the determinant of the coefficients for (26) is nonzero, the transformation ${T}_{n}$ is determined by (26) uniquely.

*λ*and

*i.e.*,

By using the above facts, we can prove the following theorem.

**Theorem 1**

*Matrices*${\tilde{U}}_{n}$

*and*${\tilde{V}}_{n}$

*determined by*(23)

*and*(24)

*have the same forms as*${U}_{n}$

*and*${V}_{n}$

*respectively*,

*where the transformation from the old potential*${M}_{n}$

*into the new one*${\tilde{M}}_{n}$

*is given by*

The proof of the form invariance for ${\tilde{U}}_{n}$, ${\tilde{V}}_{n}$ and ${U}_{n}$, ${V}_{n}$ can refer to the context in [34], the proof process is similar (for proof details, see the Appendix). According to Theorem 1, the transformations (22) and (32) can change the Lax pair (20) and (21) into the Lax pair of the same type (23) and (24). Therefore, both of Lax pairs lead to (1). Transformations (22) and (32) are called an *N*-DT of (1).

## 4 *N*-Soliton solutions and inelastic interaction of Eq. (1)

and $\mathrm{\Delta}{a}_{n}^{(-2N-1)}$ is produced from Δ by replacing its $(2N+1)$th column with ${(-{\lambda}_{1}^{2N+1},\dots ,-{\lambda}_{2N+1}^{2N+1},-{\lambda}_{1}^{-2N-1}{\delta}_{1,n},\dots ,-{\lambda}_{2N+1}^{-2N-1}{\delta}_{2N+1,n})}^{T}$, $\mathrm{\Delta}{b}_{n}^{(-2N)}$ is produced from Δ by replacing its $(4N+2)$th column with ${(-{\lambda}_{1}^{2N+1},\dots ,-{\lambda}_{2N+1}^{2N+1},-{\lambda}_{1}^{-2N-1}{\delta}_{1,n},\dots ,-{\lambda}_{2N+1}^{-2N-1}{\delta}_{2N+1,n})}^{T}$.

From (35), we can see that solution (36) is a solution in terms of determinants [38, 39]. Here we obtain the solutions in determinant form of NDDEs. However, in [42], a set of coupled conditions consisting of NDDEs is presented for Casorati determinants to solve the Toda lattice equation. The resulting set of eigenfunctions leads to complexitons through the Casoratian formulation, a feasible way has been presented to construct a broad class of Casorati determinant solutions including complexitons and generalized Casorati determinant solutions of the Toda lattice equation. Ma and a co-worker [42] also indicate that integrable equations can have three different kinds of explicit exact transcendental function solutions: negatons, positons and complexitons. Solitons are usually a specific class of negatons. Roughly speaking, negatons and positons are solutions which involve exponential functions and trigonometric functions of space variables, respectively, and they are all associated with real eigenvalues of the associated spectral problems. But complexitons are different solutions which involve both exponential and trigonometric functions of space variables, and they are associated with complex eigenvalues of the associated spectral problems [42]. It is worth pointing out that our results seem to be different from those reported in [42] considering determinant form, but Ma and a co-worker [42] pointed out that the Casorati determinant solution has actually resulted from the Darboux transformation of the Toda lattice equation. Hence we think that these solutions may be the same as Casorati determinant solutions in essence, they may be different only in form, of course, the relation between two kinds of determinant solutions is worthwhile to be studied further. However, we should point out that there are some differences between our method and [42]. Firstly, the Lax pairs are different, one is the matrix form, the other is the operator form; secondly, the deducing steps are different, comparing with [42] we directly construct the Darboux matrix ${T}_{n}$, let a Lax pair be covariant with respect to the action of the DT; thirdly, our results and Casorati determinant solutions have different forms. For our results, when choosing different *λ*, whether we can get the negatons, positons and complexitons may need further investigation. In what follows, we mainly consider multi-soliton solutions and the solitonic interaction of Eq. (1), this is the topic that we would like to address in this paper.

- (I)When $N=0$, let $\lambda ={\lambda}_{1}$. Solving the linear algebraic system (26) leads to${a}_{n}^{(-1)}=\frac{\mathrm{\Delta}{a}_{n}^{(-1)}}{\mathrm{\Delta}},\phantom{\rule{2em}{0ex}}{b}_{n}^{(0)}=\frac{\mathrm{\Delta}{b}_{n}^{(0)}}{\mathrm{\Delta}},$(37)

- (II)When $N=1$, let $\lambda ={\lambda}_{i}$ ($i=1,2,3$). Solving the linear algebraic system (26) leads to${a}_{n}^{(-3)}=\frac{\mathrm{\Delta}{a}_{n}^{(-3)}}{\mathrm{\Delta}},\phantom{\rule{2em}{0ex}}{b}_{n}^{(-2)}=\frac{\mathrm{\Delta}{b}_{n}^{(-2)}}{\mathrm{\Delta}},$(39)

With symbolic computation, solution (36) with $N=0$ and $N=1$ has been verified by substituting them into (1). When solution (36) is the soliton solution, note that solution (36) is the $(2N+1)$-soliton solution if ${\lambda}_{i}\ne 1$ and ${\lambda}_{i}\ne {\lambda}_{j}$ ($i,j=1,2,\dots ,2N+1$). However, the corresponding $(2N+1)$-soliton solution will reduce to the $(2N)$-soliton solution when one of ${\lambda}_{i}^{\prime}s$ ($i=1,2,\dots ,2N+1$, $N\ge 1$) is 1, which can be seen from Figures 2 to 3. The $(2N)$-soliton and $(2N+1)$-soliton solutions can make up the *N*-soliton solution of (1).

In [34], the elastic interaction of the solitons for a discrete system has been discussed. In this paper, we have found the inelastic interaction of the solitons in the discrete system. Therefore, we can conclude that, similar to the continuous systems, there exist the elastic interaction and inelastic interaction in the discrete systems.

## 5 Conservation laws of Eq. (1)

Conservation laws play a role in discussing the integrability for the NDDEs [34, 43], and the first three conservation laws describe the energy, momentum and Hamiltonian conservation laws, respectively. In the following, we will derive infinitely many conservation laws for (1).

*λ*in (46), we can get an infinite number of conservation laws for (1). The first two conservation laws are listed as follows:

## 6 Conclusions

In this paper, an integrable lattice hierarchy and *N*-fold DT (22) and (32) for (1) have been constructed based on its discrete spectral problem. We have derived *N*-soliton solutions (36) in terms of determinant via the resulting DT. Based on the solutions obtained, one- two- and three-solitonic structures are shown graphically: Figure 1 exhibits the one-soliton structure with $N=0$; Figures 2 and 3 show the overtaking inelastic solitonic interactions between/among the two and three solitons with $N=1$. Solitonic shapes and amplitudes have changed after the interaction. When solution (36) is solitonic, it is worth pointing out that solution (36) is the $(2N+1)$-soliton solution if ${\lambda}_{i}\ne 1$ and ${\lambda}_{i}\ne {\lambda}_{j}$ ($i,j=1,2,\dots ,2N+1$); and further, the corresponding $(2N+1)$-soliton solutions can reduce to the $(2N)$-soliton solutions if one of ${\lambda}_{i}$’s ($i=1,2,\dots ,2N+1$, $N\ge 1$) is 1. Conservation laws (47) and (48) for (1) have been explicitly given.

## Appendix

*Proof of Theorem 1*Let ${T}_{n}^{-1}={T}_{n}^{\ast}/det{T}_{n}$ and

It can be verified that ${\lambda}^{4N+2}{f}_{11}(\lambda ,n)$ is $(8N+6)$th order polynomial in *λ*, ${\lambda}^{4N+2}{f}_{12}(\lambda ,n)$ and ${\lambda}^{4N+2}{f}_{21}(\lambda ,n)$ are $(8N+5)$th order polynomials in *λ*, and ${\lambda}^{4N+2}{f}_{22}(\lambda ,n)$ is $(8N+4)$th order polynomial in *λ*.

From (23) and (54), we see that ${P}_{n}={\tilde{U}}_{n}$.

Next, we will prove that the matrix ${\tilde{V}}_{n}$ has the same form as ${V}_{n}$ under transformations (22) and (32).

It can be verified that the highest order of ${g}_{12}(\lambda ,n)$ and ${g}_{21}(\lambda ,n)$ is $4N+4$, the lowest order is $-4N-4$, and the highest and lowest orders of ${g}_{11}(\lambda ,n)$, ${g}_{22}(\lambda ,n)$ are $4N+3$ and $-4N-3$ respectively.

From (24), (60), (61) and (63), we can see that ${R}_{n}={\tilde{V}}_{n}$. The theorem is proved. □

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11201033, 91230205, 11375030.

## Authors’ Affiliations

## References

- Wen XY, Gao YT, Wang L: Darboux transformation and explicit solutions for the integrable sixth-order KdV equation for nonlinear waves.
*Appl. Math. Comput.*2011, 218: 55-60. 10.1016/j.amc.2011.05.045MathSciNetView ArticleMATHGoogle Scholar - Wadati M: Transformation theories for nonlinear discrete systems.
*Prog. Theor. Phys. Suppl.*1976, 59: 36-63.View ArticleGoogle Scholar - Ablowitz MJ, Ladik JF: On the solution of a class of nonlinear partial difference equations.
*Stud. Appl. Math.*1977, 57: 1-12.MathSciNetView ArticleMATHGoogle Scholar - Ablowitz MJ, Ladik JF: A nonlinear difference scheme and inverse scattering.
*Stud. Appl. Math.*1976, 55: 213-229.MathSciNetView ArticleMATHGoogle Scholar - Toda M:
*Theory of Nonlinear Lattices*. Springer, Berlin; 1989.View ArticleMATHGoogle Scholar - Kaup DJ: Variational solutions for the discrete nonlinear Schrödinger equation.
*Math. Comput. Simul.*2005, 69: 322-333. 10.1016/j.matcom.2005.01.015MathSciNetView ArticleMATHGoogle Scholar - Adler VE, Svinolupov SI, Yamilov RI: Multi-component Volterra and Toda type integrable equations.
*Phys. Lett. A*1999, 254: 24-36. 10.1016/S0375-9601(99)00087-0MathSciNetView ArticleMATHGoogle Scholar - Svinin AK: Reductions of the Volterra lattice.
*Phys. Lett. A*2005, 337: 197-202. 10.1016/j.physleta.2005.01.063View ArticleMATHGoogle Scholar - Zhou RG, Ma WX: Classical
*r*-matrix structures of integrable mappings related to the Volterra lattice.*Phys. Lett. A*2000, 269: 103-111. 10.1016/S0375-9601(00)00246-2MathSciNetView ArticleMATHGoogle Scholar - Zhang HW, Tu GZ, Oevel W, Fuchssteiner B: Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure.
*J. Math. Phys.*1991, 32: 1908-1918. 10.1063/1.529205MathSciNetView ArticleMATHGoogle Scholar - Ma WX, Fuchssteiner B: Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations.
*J. Math. Phys.*1999, 40: 2400-2418. 10.1063/1.532872MathSciNetView ArticleMATHGoogle Scholar - Zhang SQ: The exact solutions of a modified Volterra lattice.
*Acta Phys. Sin.*2007, 56: 1870-1874.MathSciNetMATHGoogle Scholar - Ma WX: A discrete variational identity on semi-direct sums of Lie algebras.
*J. Phys. A*2007, 40: 15055-15069. 10.1088/1751-8113/40/50/010MathSciNetView ArticleMATHGoogle Scholar - Ablowitz MJ, Segur H:
*Solitons and Inverse Scattering Transformation*. SIAM, Philadelphia; 1981.View ArticleMATHGoogle Scholar - Ablowitz MJ, Ladik JF: Nonlinear differential-difference equations.
*J. Math. Phys.*1975, 16: 598-603. 10.1063/1.522558MathSciNetView ArticleMATHGoogle Scholar - Ablowitz MJ, Clarkson PA:
*Solitons, Nonlinear Evolution Equations and Inverse Scattering*. Cambridge University Press, Cambridge; 1991.View ArticleMATHGoogle Scholar - Sun MN, Deng SF, Chen DY: The Bäcklund transformation and novel solutions for the Toda lattice.
*Chaos Solitons Fractals*2005, 23: 1169-1175. 10.1016/j.chaos.2004.06.009MathSciNetView ArticleMATHGoogle Scholar - Choudhury AG, Chowdhury AR: Canonical and Backlund transformations for discrete integrable systems and classical
*r*-matrix.*Phys. Lett. A*2001, 280: 37-44. 10.1016/S0375-9601(00)00817-3MathSciNetView ArticleMATHGoogle Scholar - Hu XB, Wu YT: Application of the Hirota bilinear formalism to a new integrable differential-difference equation.
*Phys. Lett. A*1998, 246: 523-529. 10.1016/S0375-9601(98)00571-4View ArticleGoogle Scholar - Hu XB, Ma WX: Application of Hirota’s bilinear formalism to the Toeplitz lattice some special soliton-like solutions.
*Phys. Lett. A*2002, 293: 161-165. 10.1016/S0375-9601(01)00850-7MathSciNetView ArticleMATHGoogle Scholar - Wang ZY: Darboux transformation and explicit solutions for the derivative versions of Toda equation.
*Phys. Lett. A*2008, 372: 1435-1439. 10.1016/j.physleta.2007.09.060MathSciNetView ArticleMATHGoogle Scholar - Xu XX: Darboux transformation of a coupled lattice soliton equation.
*Phys. Lett. A*2007, 362: 205-211. 10.1016/j.physleta.2006.10.014MathSciNetView ArticleMATHGoogle Scholar - Yang HX, Xu XX, Ding HY: New hierarchies of integrable positive and negative lattice models and Darboux transformation.
*Chaos Solitons Fractals*2005, 26: 1091-1103. 10.1016/j.chaos.2005.02.011MathSciNetView ArticleMATHGoogle Scholar - Ding HY, Xu XX, Zhao XD: A hierarchy of lattice soliton equations and its Darboux transformation.
*Chin. Phys.*2004, 13: 125-131. 10.1088/1009-1963/13/2/001View ArticleGoogle Scholar - Fan EG, Dai HH: A differential-difference hierarchy associated with relativistic Toda and Volterra hierarchies.
*Phys. Lett. A*2008, 372: 4578-4585. 10.1016/j.physleta.2008.04.051MathSciNetView ArticleMATHGoogle Scholar - Yang HX: Soliton solutions by Darboux transformation for a Hamiltonian lattice system.
*Phys. Lett. A*2009, 373: 741-748. 10.1016/j.physleta.2008.12.046MathSciNetView ArticleMATHGoogle Scholar - Wen XY, Gao YT: Darboux transformation and explicit solutions for discretized modified Korteweg-de Vries lattice equation.
*Commun. Theor. Phys.*2010, 53: 825-830. 10.1088/0253-6102/53/5/07MathSciNetView ArticleMATHGoogle Scholar - Gu CH, Hu HS, Zhou ZX:
*Darboux Transformation in Soliton Theory and Its Geometric Applications*. Shanghai Scientific and Technical Press, Shanghai; 1999.Google Scholar - Taogetusang , Sirendaoerji :Constructing the exact solutions of the $(2+1)$-dimensional hybrid-lattice and discrete mKdV equation.
*Acta Phys. Sin.*2007, 56: 627-636.MathSciNetMATHGoogle Scholar - Yu YX, Wang Q, Zhang HQ: New explicit rational solitary wave solutions for discretized mKdV lattice equation.
*Commun. Theor. Phys.*2005, 44: 1011-1014. 10.1088/6102/44/6/1011MathSciNetView ArticleGoogle Scholar - Zha QL, Sirendaoreji : A hyperbolic function approach to constructing exact solitary wave solutions of the hybrid lattice and discrete mKdV lattice.
*Chin. Phys.*2006, 15: 475-477. 10.1088/1009-1963/15/3/003View ArticleGoogle Scholar - Yao YQ, Zhang YF, Chen DY: Discrete integrable couplings of the Volterra lattice.
*Chin. Phys. Lett.*2007, 24: 308-311. 10.1088/0256-307X/24/2/002View ArticleGoogle Scholar - Wang Q: Travelling-wave solution of Volterra lattice by the optimal homotopy analysis method.
*Z. Naturforsch. A*2012, 67: 15-20.View ArticleGoogle Scholar - Wen XY, Gao YT:
*N*-Soliton solutions and elastic interaction of the coupled lattice soliton equations for nonlinear waves.*Appl. Math. Comput.*2012, 219: 99-107. 10.1016/j.amc.2012.04.080MathSciNetView ArticleMATHGoogle Scholar - Ablowitz MJ, Kaup DJ, Newell AC, Segur H: Nonlinear evolution equations of physical significance.
*Phys. Rev. Lett.*1973, 31: 125-127. 10.1103/PhysRevLett.31.125MathSciNetView ArticleMATHGoogle Scholar - Tu GZ: A trace identity and its applications to theory of discrete integrable systems.
*J. Phys. A*1990, 23: 3903-3922. 10.1088/0305-4470/23/17/020MathSciNetView ArticleMATHGoogle Scholar - Ma WX, Xu XX: A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations.
*J. Phys. A*2004, 37: 1323-1336. 10.1088/0305-4470/37/4/018MathSciNetView ArticleMATHGoogle Scholar - Huang DJ, Li DS, Zhang HQ: Explicit
*N*-fold Darboux transformation and multi-soliton solutions for the $(1+1)$ -dimensional higher-order Broer-Kaup system.*Chaos Solitons Fractals*2007, 33: 1677-1685. 10.1016/j.chaos.2006.03.015MathSciNetView ArticleMATHGoogle Scholar - Wang L, Gao YT, Gai XL, Yu X: Vandermonde-type odd-soliton solutions for the Whitham-Broer-Kaup model in the shallow water small-amplitude regime.
*J. Nonlinear Math. Phys.*2010, 17: 197-211. 10.1142/S1402925110000714MathSciNetView ArticleMATHGoogle Scholar - Chen AH, Li XM: Darboux transformation and soliton solutions for Boussinesq-Burgers equation.
*Chaos Solitons Fractals*2006, 27: 43-49. 10.1016/j.chaos.2004.09.116MathSciNetView ArticleMATHGoogle Scholar - Li XM, Chen AH: Darboux transformation and multi-soliton solutions of Boussinesq-Burgers equation.
*Phys. Lett. A*2005, 342: 413-420. 10.1016/j.physleta.2005.05.083MathSciNetView ArticleMATHGoogle Scholar - Ma WX, Maruno K: Complexiton solutions of the Toda lattice equation.
*Physica A*2004, 343: 219-237.MathSciNetView ArticleGoogle Scholar - Wadati M, Sanuki H, Konno K: Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws.
*Prog. Theor. Phys.*1975, 53: 419-436. 10.1143/PTP.53.419MathSciNetView ArticleMATHGoogle Scholar

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