Research Article | Open Access

# A New Integrable Variable-Coefficient -Dimensional Long Wave-Short Wave Equation and the Generalized Dressing Method

**Academic Editor:**Ben T. Nohara

#### Abstract

Based on the generalized dressing method, we propose a new integrable variable-coefficient -dimensional long wave-short wave equation and derive its Lax pair. Using separation of variables, we have derived the explicit solutions of the equation. With the aid of Matlab, the curves of the solutions are drawn.

#### 1. Introduction

It is well known that the interactions of long wave and short wave play an important role in fluid dynamics. The model is described by the following equation:The inverse scattering technique proposed in [1] plays an important role in constructing the complete solution of the long wave and short wave resonance equations. The soliton solution of long wave and short wave has been obtained in [2]. Radha et al. in [3] derived periodic solutions and localized solutions of (1). Lai and Chow in [4] studied positon and dromion solutions of -dimensional long wave and short wave resonance interaction equations. Researchers have focused on the long wave and short wave equation [5–10] by using different methods. Serkin et al. [11] discussed integrable variable-coefficient nonlinear evolution equations. By utilizing the exp-function method, the generalized solitary solution and periodic solution of soliton equations can be given in [12]. Authors in [13] discussed the interactions of dark soliton and bright soliton in a double-mode optical fiber. In [14, 15], authors (Dai and Jeffrey [14], Jeffrey and Dai [15]) extended the dressing method [16, 17] to a generalized version for solving soliton equations associated with matrix nonspectral problems and variable-coefficient cases. The generalized dressing method was based on the problems of factorization of an integral operator on the line into the product of two Volterra type integral operators , from which the Gel’fand-Levitan-Marchenko (GLM) equation is obtained. These Volterra operators are then used to construct dressed operators () starting from a pair of initial operators (). Integrable variable-coefficient nonlinear equations are obtained from the compatibility of the dressed operators. There are some differences between the original dressing method and the generalized dressing method. In the original dressing method, the constant coefficient operators have transformed into different constant coefficient operators. The generalized dressing method transforms the variable-coefficient operators into different variable-coefficient ones. The advantages in the generalized dressing method lie in deriving integrable variable-coefficient nonlinear evolution equation and corresponding Lax pairs. However, the original dressing method is a system way to study constant coefficient nonlinear evolution equation [18–20]. Authors (Dai and Jeffrey [14]) presented the generalized dressing method; we also discussed some integrable variable-coefficient evolution equations [21–25]. In fact, the dressing method can be thought as a rather general formulation of the inverse scattering method, which has the advantage of bypassing the scattering problem. The common point between the two methods is that two methods can deal with the initial boundary value problem.

In the paper, we applied the generalized dressing method to derive a new integrable variable-coefficient -dimensional long wave-short wave equation:where and are functions of and . and are functions of . Particularly, the above equation is reduced to a new -dimensional integrable variable-coefficient equation:in view of .

Furthermore, under the transformations , , (3) can be read as the cylindrical equation:Moreover, (2) are written as a -dimensional integrable modified long wave-short wave equation for and :The outline of the paper is as follows. In Section 2, we briefly describe the generalized dressing method and its properties. Moreover, we introduce two dressing operators. In Section 3, new integrable variable-coefficient -dimensional long wave-short wave equations and their Lax pairs are derived with the aid of the generalized dressing method. In Section 4, as an application, we obtain explicit solutions of these equations and draw the curves of the solutions.

#### 2. The Generalized Dressing Method and Dressing Operators

First, we consider three integral operators , , and defined by [16]We assume that exists and admits the triangular factorizationwhere is the identity operator. From (7), a direct calculation shows that and satisfy the Gel’fand-Levitan-Marchenko (GLM) equation [16]:here it is supposed that and satisfy the condition We now introduce two differential operators and defined bywith and being matrix functions of . and are matrix functions of and :Suppose that the operator commutes with and ; that is,which together with (10) implies the following equations:In what follows, we obtain the dressing operators and with the aid of operators and . The dressing procedure is accomplished through the relations [14, 15]The difference between the original dressing method and the generalized dressing method lies in the differential operators and which satisfied the relationwith and being arbitrary functions of their arguments. In view of [14, 15], the corresponding dressed operators obey the equationLetting , +, from (15a) and (15b), we haveIn view of (16), it is easy to obtainthus, we have . Here, is an arbitrary function of and . and are arbitrary functions of .

#### 3. A New Integrable Variable-Coefficient 2+1-Dimensional Long Wave-Short Wave Equation and Its Lax Pair

In this section, based on the generalized dressing method, we derive a new integrable variable-coefficient -dimensional long wave-short wave equation. From (17), we haveWe denoteIn view of (16), we haveBased on (18) and (22)-(23), we obtainFrom (20), we derive new integrable variable-coefficient -dimensional long wave-short wave equation with the aid of (22)–(26):Particularly, the above equations are reduced to the cylindrical form:where , , , and the integration constant is zero.

The Lax pairs of (27) are and . , , and are presented in (24)–(26).

Particularly, we consider the case for ; then (28) is reduced to a new coupled equation:

#### 4. Explicit Solutions and the Curves of Solutions

In this section, we shall apply the generalized dressing method to construct explicit solutions of these obtained -dimensional long wave-short wave equation and its reductions. We assume that and have solutions in the form of separation of variables:where , are some matrices.

Substituting (30) and (31) into the GLM equation (8) yields the following:where is defined by and is the Kronecker delta.

We denote , and, in view of (13) and (14), we obtainIn the what follows, we obtain one soliton solution for the case of of (30). LetThen, from (32) we obtain with

Using (22), we obtain the solutions of (27). Particularly, for , , and , we derive the solutions of (28). In what follows, we draw the curves of the solutions for , , and . Figures 1 and 2 describe the imaginary of and real of , respectively. From the curves, we can see that the forms are similar. The imaginary of and real of are shown by Figures 3 and 4, respectively. From the curves, we can see that the forms are different and with diminishing energy. Figures 5 and 6 construct the imaginary of and real of , respectively. In view of the solution curves, we can read the difference between the imaginary of and real of . Furthermore, we find that imaginary of and that of are similar. At the same time, we find that real of and that of are similar. Similarly, in later paper, we will discuss two soliton solutions and soliton solutions.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors acknowledge the support by National Natural Science Foundation of China (no. 11301149) and Henan Natural Science Foundation of Basic Research (no. 162300410072).

#### References

- Y. C. Ma, “The Complete Solution of the Long‐Wave–Short‐Wave Resonance Equations,”
*Studies in Applied Mathematics*, vol. 59, no. 3, pp. 201–221, 1978. View at: Publisher Site | Google Scholar | MathSciNet - M. Yurova, “Application of dressing method for long wave--short wave resonance interaction equation,”
*Journal of Mathematical Physics*, vol. 48, no. 5, 053516, 7 pages, 2007. View at: Publisher Site | Google Scholar | MathSciNet - R. Radha, C. Senthil Kumar, M. Lakshmanan, X. Y. Tang, and S. Y. Lou, “Periodic and localized solutions of the long wave--short wave resonance interaction equation,”
*Journal of Physics A: Mathematical and General*, vol. 38, no. 44, pp. 9649–9663, 2005. View at: Publisher Site | Google Scholar | MathSciNet - D. W. C. Lai and K. W. Chow, “"Positon" and "Dromion" Solutions of the (2+1) Dimensional Long Wave-Short Wave Resonance Interaction Equations,”
*Journal of the Physical Society of Japan*, vol. 68, no. 6, pp. 1847–1853, 1999. View at: Google Scholar - S. Y. Lou, “Symmetries and algebras of the integrable dispersive long wave equations in (2+1)-dimensional spaces,”
*Journal of Physics A: Mathematical and General*, vol. 27, no. 9, p. 3235, 1994. View at: Google Scholar - G. Symonds, D. A. Huntley, and A. J. Bowen, “Two-dimensional surf beat: long wave generation by a time-varying breakpoint,”
*Journal of Geophysical Research: Atmospheres*, vol. 87, no. 1, pp. 492–498, 1982. View at: Publisher Site | Google Scholar - R. Hirota, “Exact
*N*-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices,”*Journal of Mathematical Physics*, vol. 14, pp. 810–814, 1973. View at: Publisher Site | Google Scholar | MathSciNet - C. L. Zheng, J. P. Fang, and L. Q. Chen, “New variable separation excitations of (2+1)-dimensional dispersive long-water wave system obtained by an extended mapping approach,”
*Chaos, Solitons and Fractals*, vol. 23, no. 5, pp. 1741–1748, 2005. View at: Google Scholar - X.-Y. Tang, C.-L. Chen, and S.-Y. Lou, “Localized solutions with chaotic and fractal behaviours in a 2+1-dimensional dispersive long-wave system,”
*Journal of Physics A: Mathematical and General*, vol. 35, no. 22, pp. L293–L301, 2002. View at: Publisher Site | Google Scholar | MathSciNet - C.-L. Zheng, J.-M. Zhu, J.-F. Zhang, and L.-Q. Chen, “Fractal dromion, fractal lump, and multiple peakon excitations in a new 2+1-dimensional long dispersive wave system,”
*Communications in Theoretical Physics*, vol. 39, no. 3, pp. 261–266, 2003. View at: Publisher Site | Google Scholar | MathSciNet - V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,”
*Physical Review Letters*, vol. 98, no. 7, Article ID 074102, 2007. View at: Publisher Site | Google Scholar - S. Zhang, “Application of Exp-function method to a KdV equation with variable coefficients,”
*Physics Letters A*, vol. 365, no. 5-6, pp. 448–453, 2007. View at: Publisher Site | Google Scholar | MathSciNet - V. V. Afanasyev, Y. U. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright optical solitons,”
*Optics Expresss*, vol. 14, no. 15, pp. 805–807, 1989. View at: Publisher Site | Google Scholar - H.-H. Dai and A. Jeffrey, “The inverse scattering transforms for certain types of variable coefficient KdV equations,”
*Physics Letters A*, vol. 139, no. 8, pp. 369–372, 1989. View at: Publisher Site | Google Scholar | MathSciNet - A. Jeffrey and H. H. Dai, “On the application of a generalized version of the dressing method to the integration of variable-coefficient KdV equation,”
*Rendiconti di Matematica. Serie VII*, vol. 10, pp. 439–455, 1990. View at: Google Scholar | MathSciNet - V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,”
*Functional Analysis and its Applications*, vol. 8, no. 3, pp. 226–235, 1974. View at: Google Scholar - V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II,”
*Functional Analysis and its Applications*, vol. 13, no. 3, pp. 166–174, 1979. View at: Publisher Site | Google Scholar - A. Roy Chowdhury and S. Basak, “On the complete solution of the Hirota-Satsuma system through the “dressing'' operator technique,”
*Journal of Physics A: Mathematical and General*, vol. 17, no. 16, pp. L863–L868, 1984. View at: Publisher Site | Google Scholar | MathSciNet - J. M. Dye and A. Parker, “An inverse scattering scheme for the regularized long-wave equation,”
*Journal of Mathematical Physics*, vol. 41, no. 5, pp. 2889–2904, 2000. View at: Publisher Site | Google Scholar | MathSciNet - A. Parker, “A reformulation of the “dressing method'' for the Sawada-Kotera equation,”
*Inverse Problems*, vol. 17, no. 4, pp. 885–895, 2001. View at: Publisher Site | Google Scholar | MathSciNet - T. Su, H.-H. Dai, and X. Geng, “On the application of a generalized dressing method to the integration of variable-coefficient coupled Hirota equations,”
*Journal of Mathematical Physics*, vol. 50, no. 11, 113507, 12 pages, 2009. View at: Publisher Site | Google Scholar | MathSciNet - H. Dai and T. H. Su, “The generalized dressing method with applications to the integration of variable-coefficient Toda equations,”
*Proceedings of the Estonian Academy of Sciences*, vol. 59, no. 2, pp. 93–98, 2010. View at: Google Scholar - T. Su, H. Dai, and G. Geng X, “A Variable-Coefficient Manakov Model and Its Explicit Solutions through the Generalized Dressing Method,”
*Chinese Physics Letters*, vol. 30, no. 6, Article ID 060201, 2013. View at: Google Scholar - T. Su, H. Dai, and X. G. Geng, “On the application of a generalized version of the dressing method to the integration of variable coefficient N-coupled nonlinear Schrödinger equation,”
*Journal of Nonlinear Mathematical Physics*, vol. 19, no. 4, Article ID 1250028, 19 pages, 2012. View at: Google Scholar | MathSciNet - J. Zhu and X. Geng, “The generalized version of dressing method with applications to AKNS equations,”
*Journal of Nonlinear Mathematical Physics*, vol. 13, no. 1, pp. 81–89, 2006. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2018 Ting Su and Hui Hui Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.