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Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation
Results in Physics 17 (2020) 103031
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation
T
ARTICLE INFO
ABSTRACT
Keywords: Semi-inverse method Frequency-amplitude relationship Soliton Nonlinear vibration
Kundu–Mukherjee–Naskar equation admits many attractive solitary wave solutions as discussed by Biswas et al. (Results in Physics, Volume16, 2020, 102850). This paper shows that a variational formulation can be established by the semi-inverse method, and its periodic solution is elucidated. This short note sheds a light on new research frontiers of solitary wave theory.
Introduction
We re-write Eq. (3) in the form
This paper presents a comment on an important article by Biswas et al. [1] to lead the audience of Physics in Results to a new research frontier of the solitary wave theory and nonlinear vibration theory. The commented paper studied the following Kundu–Mukherjee–Naskar equation [1,2]
iqt + aqxy + ibq (qqx
q qx ) = 0
(1)
where a and b are constants. We assume the solution has the form
q (x , y , t ) = P ( ) exp(i (x , y , t )), =
k1 x
k2 y
= B1 x + B2 y
(2)
where B1, B2 , v, k1, k2 , and are constants. Putting Eq. (2) into Eq. (1), we obtain the real and imaginary parts:
aB1 B2 P
v=
(3)
2bk1 P 3 = 0
( + ak1 k2) P
(4)
a (k1 B2 + k2 B1)
Biswas et al. [1] made an unprecedented contribution to the solitary wave theory, and new kinds of solitary wave solutions of Eq. (1) were revealed, greatly promoting the development of solitary wave theory. Here we should attract the attention of audience of Results in Physics to the periodic solution of Eq. (1) [3]. Variational principle and periodic solution By the semi-inverse method [4–11], the variational principle can be established, which is
J (P ) =
{ 12 aB B P
where
K = 2 aB1 B2 P
1 2
2
+
}
1 1 ( + ak1 k2) P 2 + bk1 P 4 d = 2 2
{K
E}d (5)
1
2
is
the
kinetic
energy,
and
E = 2 ( + ak1 k2) P 2 2 bk1 P 4 is the potential energy, and the total energy should be kept as a constant, this implies 1
K+E=
1 aB1 B2 P 2
2
1
1 ( + ak1 k2) P 2 2
Here H is the Hamilton constant.
1 bk1 P 4 = H 2
( + ak1 k2) P aB1 B2
2bk1 3 P =0 aB1 B2
(7)
By He’s frequency formulation [12–16], its periodic solution is
P ( ) = A cos(
+
(8)
)
where A is the amplitude and
=
( + ak1 k2) aB1 B2
3 2bk1 2 A 4 aB1 B2
> 0.
Conclusion
vt ,
t+
P
(6)
This short note shows a nonlinear equation might admit both solitary wave solutions and periodic solutions. That means a nonlinear wave equation might have some periodic properties, and a nonlinear vibration equation might have solitary properties. This finding can greatly widen our sight and richen our knowledge on solitary wave theory and nonlinear vibration theory. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Biswas A, Vega-Guzman J, Bansal A, et al. Optical dromions, domain walls and conservation laws with Kundu–Mukherjee–Naskar equation via traveling waves and Lie symmetry. Results Phys 2020;16:102850. [2] Ekici M, Sonmezoglu A, Biswas A, et al. Optical solitons in (2+1)-Dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme. Chin J Phys 2019;57:72–7. [3] Ji FY, He CH, Zhang JJ, et al. A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar. Appl Math Model 2020;82:437–48. https://doi.org/10.1016/j.apm.2020.01.027. [4] He JH. Generalized variational principles for buckling analysis of circular cylinders. Acta Mech 2019. https://doi.org/10.1007/s00707-019-02569-7. [5] He JH. A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals https://doi.org/10.1142/S0218348X20500243. [6] He JH. A modified Li-He’s variational principle for plasma. Int J Numer Meth Heat Fluid Flow 2019. https://doi.org/10.1108/HFF-06-2019-0523. [7] He JH. Lagrange crisis and generalized variational principle for 3D unsteady flow. Int J Numer Meth Heat Fluid Flow 2019. https://doi.org/10.1108/HFF-07-2019057.
https://doi.org/10.1016/j.rinp.2020.103031 Received 15 February 2020; Received in revised form 26 February 2020; Accepted 26 February 2020 Available online 28 February 2020 2211-3797/ © 2020 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).
Results in Physics 17 (2020) 103031 [8] He CH, Shen Y, Ji FY, et al. Taylor series solution for fractal Bratu-type equation arising in electrospinning process. Fractals 2020;28(1):2050011. doi: 10.1142/ S0218348X20500115. [9] He JH, Latifizadeh H. A general numerical algorithm for nonlinear differential equations by the variational iteration method. Int J Numer Meth Heat Fluid Flow 2020. https://doi.org/10.1108/HFF-01-2020-0029. [10] He JH. A short review on analytical methods for to a fully fourth-order nonlinear integral boundary value problem with fractal derivatives. Int J Numer Meth Heat Fluid Flow 2020. https://doi.org/10.1108/HFF-01-2020-0060. [11] He Ji-Huan, Ain Qura-Tul. New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle. Therm sci 2020:65. https://doi.org/10.2298/TSCI200127065H. [12] He JH. The simplest approach to nonlinear oscillators. Results Phys 2019;15:102546https://doi.org/10.1016/j.rinp.2019.102546. [13] He JH. The simpler, the better: analytical methods for nonlinear oscillators and fractional oscillators. J Low Freq Noise Vibr Active Control 2019;38:1252–60. [14] He JH, Jin X. A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube. Mathemat Methods Appl Sci 2020. https://doi.org/10.
1002/mma.6321. [15] Li XX, He JH. Nanoscale adhesion and attachment oscillation under the geometric potential. Part 1: The formation mechanism of nanofiber membrane in the electrospinning. Results Phys 2019;12:1405–10. [16] Li XX, Li YY, Li Y, et al. Gecko-like adhesion in the electrospinning process. Results Phys 2020;16:102899.
a
2
Ji-Huan Hea,b,c School of Science, Xi'an University of Architecture and Technology, Xi’an, China b School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China c National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, Suzhou, China E-mail address: [email protected].
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation
T
ARTICLE INFO
ABSTRACT
Keywords: Semi-inverse method Frequency-amplitude relationship Soliton Nonlinear vibration
Kundu–Mukherjee–Naskar equation admits many attractive solitary wave solutions as discussed by Biswas et al. (Results in Physics, Volume16, 2020, 102850). This paper shows that a variational formulation can be established by the semi-inverse method, and its periodic solution is elucidated. This short note sheds a light on new research frontiers of solitary wave theory.
Introduction
We re-write Eq. (3) in the form
This paper presents a comment on an important article by Biswas et al. [1] to lead the audience of Physics in Results to a new research frontier of the solitary wave theory and nonlinear vibration theory. The commented paper studied the following Kundu–Mukherjee–Naskar equation [1,2]
iqt + aqxy + ibq (qqx
q qx ) = 0
(1)
where a and b are constants. We assume the solution has the form
q (x , y , t ) = P ( ) exp(i (x , y , t )), =
k1 x
k2 y
= B1 x + B2 y
(2)
where B1, B2 , v, k1, k2 , and are constants. Putting Eq. (2) into Eq. (1), we obtain the real and imaginary parts:
aB1 B2 P
v=
(3)
2bk1 P 3 = 0
( + ak1 k2) P
(4)
a (k1 B2 + k2 B1)
Biswas et al. [1] made an unprecedented contribution to the solitary wave theory, and new kinds of solitary wave solutions of Eq. (1) were revealed, greatly promoting the development of solitary wave theory. Here we should attract the attention of audience of Results in Physics to the periodic solution of Eq. (1) [3]. Variational principle and periodic solution By the semi-inverse method [4–11], the variational principle can be established, which is
J (P ) =
{ 12 aB B P
where
K = 2 aB1 B2 P
1 2
2
+
}
1 1 ( + ak1 k2) P 2 + bk1 P 4 d = 2 2
{K
E}d (5)
1
2
is
the
kinetic
energy,
and
E = 2 ( + ak1 k2) P 2 2 bk1 P 4 is the potential energy, and the total energy should be kept as a constant, this implies 1
K+E=
1 aB1 B2 P 2
2
1
1 ( + ak1 k2) P 2 2
Here H is the Hamilton constant.
1 bk1 P 4 = H 2
( + ak1 k2) P aB1 B2
2bk1 3 P =0 aB1 B2
(7)
By He’s frequency formulation [12–16], its periodic solution is
P ( ) = A cos(
+
(8)
)
where A is the amplitude and
=
( + ak1 k2) aB1 B2
3 2bk1 2 A 4 aB1 B2
> 0.
Conclusion
vt ,
t+
P
(6)
This short note shows a nonlinear equation might admit both solitary wave solutions and periodic solutions. That means a nonlinear wave equation might have some periodic properties, and a nonlinear vibration equation might have solitary properties. This finding can greatly widen our sight and richen our knowledge on solitary wave theory and nonlinear vibration theory. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Biswas A, Vega-Guzman J, Bansal A, et al. Optical dromions, domain walls and conservation laws with Kundu–Mukherjee–Naskar equation via traveling waves and Lie symmetry. Results Phys 2020;16:102850. [2] Ekici M, Sonmezoglu A, Biswas A, et al. Optical solitons in (2+1)-Dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme. Chin J Phys 2019;57:72–7. [3] Ji FY, He CH, Zhang JJ, et al. A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar. Appl Math Model 2020;82:437–48. https://doi.org/10.1016/j.apm.2020.01.027. [4] He JH. Generalized variational principles for buckling analysis of circular cylinders. Acta Mech 2019. https://doi.org/10.1007/s00707-019-02569-7. [5] He JH. A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals https://doi.org/10.1142/S0218348X20500243. [6] He JH. A modified Li-He’s variational principle for plasma. Int J Numer Meth Heat Fluid Flow 2019. https://doi.org/10.1108/HFF-06-2019-0523. [7] He JH. Lagrange crisis and generalized variational principle for 3D unsteady flow. Int J Numer Meth Heat Fluid Flow 2019. https://doi.org/10.1108/HFF-07-2019057.
https://doi.org/10.1016/j.rinp.2020.103031 Received 15 February 2020; Received in revised form 26 February 2020; Accepted 26 February 2020 Available online 28 February 2020 2211-3797/ © 2020 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).
Results in Physics 17 (2020) 103031 [8] He CH, Shen Y, Ji FY, et al. Taylor series solution for fractal Bratu-type equation arising in electrospinning process. Fractals 2020;28(1):2050011. doi: 10.1142/ S0218348X20500115. [9] He JH, Latifizadeh H. A general numerical algorithm for nonlinear differential equations by the variational iteration method. Int J Numer Meth Heat Fluid Flow 2020. https://doi.org/10.1108/HFF-01-2020-0029. [10] He JH. A short review on analytical methods for to a fully fourth-order nonlinear integral boundary value problem with fractal derivatives. Int J Numer Meth Heat Fluid Flow 2020. https://doi.org/10.1108/HFF-01-2020-0060. [11] He Ji-Huan, Ain Qura-Tul. New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle. Therm sci 2020:65. https://doi.org/10.2298/TSCI200127065H. [12] He JH. The simplest approach to nonlinear oscillators. Results Phys 2019;15:102546https://doi.org/10.1016/j.rinp.2019.102546. [13] He JH. The simpler, the better: analytical methods for nonlinear oscillators and fractional oscillators. J Low Freq Noise Vibr Active Control 2019;38:1252–60. [14] He JH, Jin X. A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube. Mathemat Methods Appl Sci 2020. https://doi.org/10.
1002/mma.6321. [15] Li XX, He JH. Nanoscale adhesion and attachment oscillation under the geometric potential. Part 1: The formation mechanism of nanofiber membrane in the electrospinning. Results Phys 2019;12:1405–10. [16] Li XX, Li YY, Li Y, et al. Gecko-like adhesion in the electrospinning process. Results Phys 2020;16:102899.
a
2
Ji-Huan Hea,b,c School of Science, Xi'an University of Architecture and Technology, Xi’an, China b School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China c National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, Suzhou, China E-mail address: [email protected].