# Integrability via Functional Expansion for the KMN Model

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The KMN Model

**Remark**

**1.**

## 3. Auxiliary Equations

**Example**

**1.**

**Example**

**2.**

## 4. The Functional Expansion Method

## 5. Solving the KMN Equation

#### 5.1. Functional Expansion with a First Order Auxiliary Equation

#### 5.2. Functional Expansion with a Second Order Auxiliary Equation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett.
**1973**, 31, 125–127. [Google Scholar] [CrossRef] - Hirota, R. The Direct Method in Soliton Theory; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Olver, P.J. Applications of Lie groups to differential equations. In Graduate Texts in Mathematics; Springer: New York, NY, USA, 1993; p. 10. ISBN 978-0-387-95000-6. [Google Scholar]
- Cimpoiasu, R.; Constantinescu, R.; Cimpoiasu, V.M. Integrability of dynamical systems with polynomial Hamiltonians. Rom. J. Phys.
**2005**, 50, 317–324. [Google Scholar] - Cimpoiasu, R.; Constantinescu, R. Lie symmetries for Hamiltonian systems methodological approach. Int. J. Theor. Phys.
**2006**, 45, 1769–1782. [Google Scholar] [CrossRef] - Malfliet, W. Solitary Wave Solutions of Nonlinear Wave Equations. Am. J. Phys.
**1992**, 60, 650–654. [Google Scholar] [CrossRef] - Wazwaz, A.M. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput.
**2007**, 188, 1467–1475. [Google Scholar] [CrossRef] - Wang, M.L.; Li, X.Z. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos Soliton Fract.
**2005**, 24, 1257–1268. [Google Scholar] [CrossRef] - He, J.H.; Wu, X.H. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals
**2006**, 30, 700–708. [Google Scholar] [CrossRef] - Liu, S.; Fu, Z.; Liu, S.; Zhao, Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A
**2001**, 289, 69–74. [Google Scholar] [CrossRef] - Wang, M.; Li, X.; Zhang, J. The (G
^{′}/G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Phys. Lett A**2008**, 372, 417–423. [Google Scholar] [CrossRef] - Zhang, J.; Jiang, F.; Zhao, X. An improved (G′/G)-expansion method for solving nonlinear evolution equations. Int. J. Comput. Math.
**2010**, 87, 1716–1725. [Google Scholar] [CrossRef] - Akbar, M.A.; Ali, N.H.M.; Zayed, E.M.E. A generalized and improved (G
^{′}/G)-expansion method for nonlinear evolution equations. Math. Prob. Eng.**2012**, 2012, 22. [Google Scholar] [CrossRef][Green Version] - Alam, M.N.; Belgacem, F.B.M. Exact Traveling Wave Solutions for the (1 + 1)-Dimensional Compound KdVB Equation via the Novel (G′/G)-Expansion Method. Intern. J. Mod. Nonlin. Theory Appl.
**2016**, 5, 28. [Google Scholar] [CrossRef][Green Version] - Constantinescu, R.; Ionescu, C.; Stoicescu, M. Functional expansions for finding traveling wave solutions. J. Appl. Anal. Comput.
**2020**, 10, 569–583. [Google Scholar] - Rezazadeh, H.; Korkmaz, A.; Eslami, M.; Mirhosseini-Alizamini, S.M. A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method. Opt. Quant. Electron.
**2019**, 51, 84. [Google Scholar] [CrossRef] - Cimpoiasu, R.; Pauna, A.S. Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method. Open Phys. J.
**2018**, 16, 419–426. [Google Scholar] [CrossRef] - Khater, M.; Alzaidi, J.F.; Attia, R.A.; Lu, D. Analytical and numerical solutions for the current and voltage model on an electrical transmission line with time and distance. Phys. Scr.
**2019**, 95, 055206. [Google Scholar] [CrossRef] - Weiss, J. The Painleve property for partial differential equations. II: Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys.
**1983**, 24, 1405. [Google Scholar] [CrossRef] - Lu, D.; Seadawy, A.R.; Khater, M.M. Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system. Adv. Differ. Equ.
**2018**, 1, 266. [Google Scholar] [CrossRef][Green Version] - Kundu, A.; Mukherjee, A. Novel integrable higher-dimensional nonlinear Schrődinger equation: Properties, solutions, applications. arXiv
**2013**, arXiv:1305.4023. [Google Scholar] - Ekici, M.; Sonmezoglu, A.; Biswas, A.; Belic, M.R. Optical solitons in (2+ 1)-Dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme. Chin. J. Phys.
**2019**, 57, 72–77. [Google Scholar] [CrossRef] - Yildirim, Y. Optical solitons to Kundu-Mukherjee-Naskar model with modified simple equation approach. Optik
**2019**, 184, 247–252. [Google Scholar] [CrossRef] - Jhangeer, A.; Seadawy, A.R.; Ali, F.; Ahmed, A. New complex waves of perturbed Shrődinger equation with Kerr law nonlinearity and Kundu-Mukherjee-Naskar equation. Results Phys.
**2020**, 16, 102816. [Google Scholar] [CrossRef] - Aliyu, A.I.; Li, Y.; Baleanu, D. Single and combined optical solitons, and conservation laws in (2 + 1)-dimensions with Kundu-Mukherjee-Naskar equation. Chin. J. Phys.
**2020**, 63, 410–418. [Google Scholar] [CrossRef] - Sulaiman, T.A.; Bulut, H. The new extended rational SGEEM for construction of optical solitons to the (2 + 1)-dimensional Kundu-Mukherjee-Naskar model. Appl. Math. Nonlin.Sci.
**2019**, 4, 513–522. [Google Scholar] [CrossRef][Green Version] - Wen, X. Higher-order rational solutions for the (2+1)-dimensional KMN equation. Proc. Rom. Acad. A.
**2017**, 18, 191–198. [Google Scholar] - Kundu, A.; Mukherjee, A.; Naskar, T. Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents. Proc. Math Phys. Eng. Sci.
**2017**, 470, 20130576. [Google Scholar] [CrossRef] - Zhang, S.; Wang, J.; Peng, A.; Bin, C. A generalized exp-function method for multiwave solutions of sine-Gordon equation. Pramana J. Phys.
**2013**, 81, 763–773. [Google Scholar] [CrossRef] - Singh, S.; Mukherjee, A.; Sakkaravarthi, K.; Murugesan, K. Higher dimensional localized and periodic wave dynamics in a new integrable (2+1)-dimensional Kundu-Mukherjee-Naskar mode. arXiv
**2020**, arXiv:2001.06766v1. [Google Scholar] - Khater, M.M. Extended Exp (-Î$\frac{3}{4}$)-Expansion Method for Solving the Generalized Hirota-Satsuma Coupled KdV System. Glob. J. Sci. Front. Res.
**2015**, 15, 1. [Google Scholar] - Cimpoiasu, R. Travelling wave solutions for the Long-Short wave resonance model through an improved (G
^{′}/G)-expansion method. Rom. J. Phys.**2018**, 63, 111. [Google Scholar] - Cimpoiasu, R.; Cimpoiasu, V.; Constantinescu, R. Nonlinear dynamical systems in various space-time dimensions. Rom. J. Phys.
**2010**, 55, 25–35. [Google Scholar] - Cimpoiasu, R. Integrability features for the abelian gauge field. Rom. Rep. Phys.
**2005**, 57, 167. [Google Scholar] - Kudryashov, N.A. Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci.
**2009**, 14, 3507–3529. [Google Scholar] [CrossRef][Green Version] - Cimpoiasu, R.; Constantinescu, R. Nonlinear self-adjointness and invariant solutions of a 2D Rossby wave equation. Cent. Eur. J. Phys.
**2014**, 12, 81–89. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The graphical representations corresponding to the real part of the solution $q\left(\right)open="("\; close=")">t,x,0$ from (46) with the parametric relations $\beta =1/2=-{b}_{1}=-{b}_{2},\alpha =\omega ={k}_{1}={k}_{2}=1:$(

**a**) the surface plot; (

**b**) the contour plot of (

**a**).

**Figure 2.**The graphical representations corresponding to the real part of the solution $q\left(\right)open="("\; close=")">t,x,0$ from (50) with the parametric choices $\beta ={b}_{1}={b}_{2}-\omega /4=1/2,;{k}_{1}={k}_{2}=\alpha =1:$ (

**a**) the surface plot and (

**b**) the contour plot of (

**a**).

**Figure 3.**Multi-solitons corresponding to the real part of the solution $q\left(\right)open="("\; close=")">t,x,0$ from (61) with the parametric choices ${k}_{1}={k}_{2}=1,\beta =-{b}_{1}=-{b}_{2}=1/2,\omega =1,\alpha =1(v=1)$: (

**a**) the surface plot with a grid [50,50]. (

**b**) the surface plot with a grid [1000,1000], and (

**c**) the contour plot of (

**a**).

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Constantinescu, R.; Florian, A.
Integrability via Functional Expansion for the KMN Model. *Symmetry* **2020**, *12*, 1819.
https://doi.org/10.3390/sym12111819

**AMA Style**

Constantinescu R, Florian A.
Integrability via Functional Expansion for the KMN Model. *Symmetry*. 2020; 12(11):1819.
https://doi.org/10.3390/sym12111819

**Chicago/Turabian Style**

Constantinescu, Radu, and Aurelia Florian.
2020. "Integrability via Functional Expansion for the KMN Model" *Symmetry* 12, no. 11: 1819.
https://doi.org/10.3390/sym12111819