Interactions of solitons for a generalized nonlinear Schrödinger equation from the inhomogeneous Heisenberg ferromagnetic spin system

Journal Pre-proof Interactions of solitons for a generalized nonlinear Schr¨odinger equation from the inhomogeneous Heisenberg ferromagnetic spin system

Pan Wang, Yan-Li Li, Feng-Hua Qi

PII: DOI: Reference:

S0893-9659(19)30463-X https://doi.org/10.1016/j.aml.2019.106139 AML 106139

To appear in:

Applied Mathematics Letters

Received date : 6 October 2019 Revised date : 11 November 2019 Accepted date : 11 November 2019 Please cite this article as: P. Wang, Y.-L. Li and F.-H. Qi, Interactions of solitons for a generalized nonlinear Schr¨odinger equation from the inhomogeneous Heisenberg ferromagnetic spin system, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106139. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

Journal Pre-proof

Interactions of solitons for a generalized nonlinear Schr¨odinger equation from the inhomogeneous Heisenberg ferromagnetic spin system Pan Wang1∗, Yan-Li Li1 and Feng-Hua Qi2 1 Sports Business School, Beijing Sport University, Information Road Haidian District, Beijing, No.48, 100084, China School of Information, Beijing Wuzi University, Beijing 101149, China

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2

Abstract

In this paper, we investigate the following inhomogeneous generalized nonlinear Schr¨ odinger(NLS) equation, generated by deforming the inhomogeneous Heisenberg ferromagnetic spin system through the space curve formalism and using the prolongation structure theory. Bilinear form and two-soliton solutions for the inhomogeneous generalized NLS equation are obtained. Infinitely many conservation laws for the inhomogeneous generalized NLS equation are also derived. Interactions of solitons and properties of propagation are investigated analytically and graphically, which are different from the types with the only time variables or the single space variables.

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PACS numbers: 05.45.Yv, 02.30.Ik, 02.30.Jr, 02.70.Wz Keywords: Inhomogeneous generalized nonlinear Schr¨ odinger equation; Infinitely many conversation laws; Hirota method

∗

Corresponding author, with e-mail address as wang− [email protected]

1

Journal Pre-proof 1. Introduction Nonlinear evolution equations (NLEEs) used to describe the nonlinear phenomena, have the attractive applications in such fields as condensed matter physics, fluids and particle physics [1–6]. It should be mentioned that those integrable NLEEs with constant coefficients are regarded to be idealized in physical situations [7]. Hence, the realistic description of physical phenomena should take into account the inhomogeneities in the medium [8]. The problem of nonlinear wave propagation in the dispersive and inhomogeneous media has attracted considerable interest and has some applications, i.e. waves in the ocean, radio waves in the ionosphere, nonlinear optics and magnetic systems [9–11]. In this paper, we consider the following inhomogeneous generalized nonlinear Schrodinger(NLS) equation [12, 13] generated by deforming the inhomogeneous Heisenberg ferromagnetic spin system through the space curve formalism and using the prolongation structure theory: ¾ ½ Z x 2 2 2 i ψt + i ² ψxxx + 6 i ² |ψ| ψx + (f ψ)xx + 2 ψ f |ψ| + fx |ψ| ds − i (h ψ)x = 0.

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−∞

(1)

Here, the function ψ = ψ(x, t) is a complex variable, f and h represent the inhomogeneous present in the medium with the form f = µ1 (t) x + ν1 (t) and h = µ2 (t) x + ν2 (t), µ1 (t), µ2 (t), ν1 (t) and ν2 (t) are functions of the variable t, ² is the parameter. ∗ denotes the complex conjunction. Lax pair for Eq. (1) has given in Ref. [12, 14, 15]. It should be mentioned that f and h are the functions of x and t. For the case of f = µ1 (t) x + ν1 (t) and h = µ2 (t) x + ν2 (t), the integrable properties and soliton solutions for Eq. (1) have not been investigated. Motivated by the above considerations, in this paper, we aim at deriving the integrable properties and soliton solutions, analyzing the interactions of solitons and properties of propagation, and discussing the effect of the parameters µ1 (t), µ2 (t), ν1 (t) and ν2 (t) on the soliton for Eq. (1). The plan of the paper will be as follows: The bilinear form and soliton solutions for Eq. (1) will be obtained in Section 2. In Section 3, infinitely many conservation laws will be derived through the Lax pair for Eq. (1). In Section 4, interaction of solitons and effect of the parameters µ1 (t), µ2 (t), ν1 (t) and ν2 (t) on the soliton will be studied graphically and analytically. Results will be concluded in Section 5.

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2. Soliton solutions for Eq. (1)

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2.1. Bilinear Form With the help of symbolic computation, we can derive the bilinear form for Eq. (1) as

2

Journal Pre-proof follows:

½

¾ i h i 2 3 i Dt + − i ν2 (t) − i µ2 (t) x + 2 µ1 (t) Dx + µ1 (t) x + ν1 (t) Dx + i ² Dx g · s h

−i µ2 (t) g s + 2 µ1 (t) g sx = 0,

Dx2 s

(2a)

∗

· s − 2 g g = 0.

(2b)

Hereby, Dx and Dt are the bilinear derivative operators [16] defined by µ ¶m µ ¶n £ ¤ ∂ ∂ ∂ ∂ m n Dx Dt µ(x, t) · ν(x, t) = µ(x, t) ν(x0 , t0 ) − 0 − 0 ∂x ∂x ∂t ∂t

¯ ¯ ¯ ¯

, x0 =x, t0 =t

where m and n are the nonnegative integers. Here, q = gs , g(x, t) is a complex differentiable function, s(x, t) is a real one, the parameters f = µ1 (t) x + ν1 (t), h = µ2 (t) x + ν2 (t).

∗

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2.2. Soliton Solutions After some calculations, the two-soliton solution for Eq. (1) can be expressed as g1 + g3 g = g1 + g3 , s = 1 + s2 + s4 , g1 = eξ1 + eξ2 , ψ= 1 + s2 + s4 R e µ2 (t)dt R R ϕj (t) = , ξj = ϕj (t) x + φj (t), (j = 1, 2), cj − i µ1 (t) e µ2 (t)dt dt Z n o φj (t) = − ² ϕj (t)3 + i ν1 (t) ϕj (t)2 + µ2 (t) + ϕj (t)[2 i µ1 (t) + ν2 (t)] dt (j = 1, 2), ∗

g3 = ρ1 (t) eξ1 +ξ2 +ξ1 + ρ2 (t) eξ1 +ξ2 +ξ2 , ∗

∗

∗

s4 = %5 (t) eξ1 +ξ2 +ξ1 +ξ2 ,

∗

∗

∗

s2 = %1 (t) eξ1 +ξ1 + %2 (t) eξ1 +ξ2 + %3 (t) eξ2 +ξ1 + %4 (t) eξ2 +ξ2 , R R R e−2 µ2 (t)dt (c1 − c2 )2 [c∗1 + i µ1 (t)e µ2 (t)dt dt]4 ρ1 (t) = , (c1 + c∗1 )2 (c2 + c∗1 )2 R R R e−2 µ2 (t)dt (c1 − c2 )2 [c∗2 + i µ1 (t)e µ2 (t)dt dt]4 , ρ2 (t) = (c1 + c∗2 )2 (c2 + c∗2 )2 R R R R R e−2 µ2 (t)dt [i c1 + µ1 (t) e µ2 (t)dt ]2 [−i c∗1 + µ1 (t) e µ2 (t)dt ]2 %1 (t) = , (c1 + c∗1 )2 R R R R R e−2 µ2 (t)dt [i c1 + µ1 (t) e µ2 (t)dt ]2 [−i c∗2 + µ1 (t) e µ2 (t)dt ]2 %2 (t) = , (c1 + c∗2 )2

R R R µ1 (t) e µ2 (t)dt ]2 [−i c∗1 + µ1 (t) e µ2 (t)dt ]2 %3 (t) = , (c2 + c∗1 )2 R R R R R e−2 µ2 (t)dt [i c2 + µ1 (t) e µ2 (t)dt ]2 [−i c∗2 + µ1 (t) e µ2 (t)dt ]2 %4 (t) = , (c2 + c∗2 )2 R

µ2 (t)dt

[i c2 +

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e−2

R

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R

%5 (t) = e−4 µ2 (t)dt (c1 − c2 )2 (c∗1 − c∗2 )2 (c1 + c∗1 )−2 (c1 + c∗2 )−2 (c2 + c∗1 )−2 (c2 + c∗2 )−2 Z Z R R µ2 (t)dt 2 ×[c1 − i µ1 (t) e dt] [c2 − i µ1 (t) e µ2 (t)dt dt]2 Z Z R R ∗ µ2 (t)dt 2 ∗ ×[c1 + i µ1 (t) e dt] [c2 + i µ1 (t) e µ2 (t)dt dt]2 , (3) 3

Journal Pre-proof with c1 and c2 as the complex constants. 3. Infinitely many conservation laws for Eq. (1) Based on the linear system [14–17], the infinitely many conservation laws for Eq. (1) can be expressed as follows: ∂ ∂ Rj + Jj = 0 (j = 1, 2, 3, . . .), ∂t ∂x

(4)

with 1 R1 = − i ψ ψ ∗ , 2 in J1 = − µ2 (t) ψ ψ ∗ x − i µ1 (t) ψx ψ ∗ x + i µ1 (t) ψ ψx∗ x − ν2 (t) ψ ψ ∗ + 3 ² ψ 2 (ψ ∗ )2 2 o −i ν1 (t) ψx ψ ∗ + i ν1 (t) ψ [ψ ∗ ]x − ² ψx [ψ ∗ ]x + ² ψxx ψ ∗ + ² ψ [ψ ∗ ]xx − i µ1 (t) ψ ψ ∗ ,

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1 R2 = − ψ (ψ ∗ )x , 4 in J2 = µ1 (t) ψ 2 [ψ ∗ ]2 x + ν1 (t) ψ 2 [ψ ∗ ]2 − µ1 (t) ψ [ψ ∗ ]x + i µ2 (t) ψ [ψ ∗ ]x x + i ν2 (t) ψ [ψ ∗ ]x 4 −6 i ² ψ 2 ψ ∗ [ψ ∗ ]x − µ1 (t) ψx [ψ ∗ ]x x − ν1 (t) ψx [ψ ∗ ]x − i ² ψxx [ψ ∗ ]x + µ1 (t) ψ [ψ ∗ ]xx x o ∗ ∗ ∗ +ν1 (t) ψ [ψ ]xx + i ² ψx [ψ ]xx − i ² ψ [ψ ]xxx , i 1 h 2 ∗2 ∗ R3 = i ψ ψ + ψ (ψ )xx , 8 in J3 = − − i µ1 (t) ψ 2 [ψ ∗ ]2 − µ2 (t) ψ 2 [ψ ∗ ]2 x − ν2 (t) ψ 2 [ψ ∗ ]2 + 4 ² ψ 3 [ψ ∗ ]3 8 −² [ψ ∗ ]2 [ψx ]2 + 4 i µ1 (t) ψ 2 ψ ∗ [ψ ∗ ]x x + 4 i ν1 (t) ψ 2 ψ ∗ [ψ ∗ ]x + 2 ² ψ ψ ∗ ψx [ψ ∗ ]x +5 ² ψ 2 [(ψ ∗ )x ]2 + 2 ² ψ [ψ ∗ ]2 ψxx − i µ1 (t) ψ [ψ ∗ ]xx − µ2 (t) ψ [ψ ∗ ]xx x − ν2 (t) ψ [ψ ∗ ]xx +8 ² ψ 2 ψ ∗ [ψ ∗ ]xx − i µ1 (t) ψx [ψ ∗ ]xx x − i ν1 (t) ψx [ψ ∗ ]xx + ² ψxx [ψ ∗ ]xx o +i µ1 (t) ψ [ψ ∗ ]xxx x + i ν1 (t) ψ [ψ ∗ ]xxx − ² ψx [ψ ∗ ]xxx + ² ψ [ψ ∗ ]xxxx , .. .

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where Rj ’s and Jj ’s (j = 1, 2, 3, · · · ) denote the conserved densities and conserved fluxes, respectively.

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4. Analysis and Discussions One-soliton solution for Eq. (1) can be given as ψ = g1 /(1 + s2 ), g1 = eϕ(t) x+φ(t) ½ , R R ∗ R ξ+ξ −2 µ (t)dt µ2 (t)dt R s2 = (c+c∗ )2 [c−i R µ (t) eRe µ2 (t)dt dt]22[c∗ +i R µ (t) eR µ2 (t)dt dt]2 , ϕ(t) = c−i R µe (t)e − µ2 (t)dt dt , φ(t) = 1 1 1 ¾ h i ² ϕ(t)3 + i ϕ(t)2 ν1 (t) + µ2 (t) + ϕ(t) 2 i µ1 (t) + ν2 (t) dt. 4

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0.07 ÈΨÈ

4

0.2

2

ÈΨÈ2 0.4

2

ÈΨÈ

0

0

12

3

2 x

-10 x

t 24

(a)

x

t 0

-1

1

1

-1

t 17

(b)

0.4

(c)

Propagation of one soliton via the one-soliton solution. Parameters are (a) ² = 1, µ1 (t) = −2,

Figs. 1.

ν1 (t) = −2, µ2 (t) = 1, ν2 (t) = 1, c = 3; (b) ² = 1, µ1 (t) = 1, ν1 (t) = 2, µ2 (t) = −1, ν2 (t) = t, c = 1; (c)

0.3

0.3

ÈΨÈ2

2

ÈΨÈ

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² = 1, µ1 (t) = 1, ν1 (t) = 1, µ2 (t) = 4, ν2 (t) = 2, c = 3.

0 -2

2

x

0.3

0 -17

-2

(a)

0

x

t

7

ÈΨÈ2

0 -17 x

t

7

0 t 7

-2

(b)

-2

(c)

Interaction of two solitons via Solutions (3). Parameters are (a) ² = 1, µ1 (t) = −1, ν1 (t) = −2,

Figs. 2.

µ2 (t) = 1, ν2 (t) = 1, c1 = 3, c2 = 2; (b) ² = 1, µ1 (t) = −1, ν1 (t) = 1, µ2 (t) = −2, ν2 (t) = −2, c1 = 3, c2 = −2; (c) ² = 1, µ1 (t) = 1, ν1 (t) = −1, µ2 (t) = −2, ν2 (t) = 2, c1 = 3, c2 = −2.

10

0.2

0.15

ÈΨÈ2

2

ÈΨÈ

0 -7

2

x

t

(a)

0

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17

Figs. 3.

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ÈΨÈ2

0 -6

1

x

t -1

16

0 -12

1 x

-1 t

6

(b)

(c)

Interaction of two solitons via Solutions (3). Parameters are (a) ² = 1, µ1 (t) = −1, ν1 (t) = 1,

µ2 (t) = 2, ν2 (t) = −2, c1 = 3, c2 = −2; (b)² = 1, µ1 (t) = 1, ν1 (t) = t, µ2 (t) = −1, ν2 (t) = 2 t, c1 = −2, c2 = 1; (c)² = 1, µ1 (t) = 1, ν1 (t) = −sin(t), µ2 (t) = −1, ν2 (t) = sin(0.5 t), c1 = −2, c2 = 1.

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Journal Pre-proof

0.8

0.6

0.8 ÈΨÈ

ÈΨÈ

0 -20

2 x

t 20

ÈΨÈ

0 -10

2 x

t

3

x

t

-2

(a) Figs. 4.

-18

3

20

-2

0

(b)

-2

(c)

Interaction of two solitons via Solutions (3). Parameters are (a) ² = 1, µ1 (t) = 1, ν1 (t) = sin(t),

µ2 (t) = −1, ν2 (t) = t, c1 = −2, c2 = 1; (b)² = 1, µ1 (t) = 1, ν1 (t) = sin(t), µ2 (t) = 1, ν2 (t) = t3 , c1 = −2, c2 = 1; (c)² = 1, µ1 (t) = 1, ν1 (t) = sin(t), µ2 (t) = 1, ν2 (t) = cos(0.5 t), c1 = −2, c2 = 1.

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In Fig. 1(a), at the beginning, the amplitude of the soliton increases and the velocity decreases. When the velocity of the soliton comes to zero, the direction of the velocity changes, and then, the amplitude of the soliton decreases and the velocity of the soliton increases. During the whole process, as the amplitude of the soliton reaches the maximum value, the velocity of the soliton attains the minimum one. The amplitude of the soliton increases suddenly, and it decreases gradually; the velocity of the soliton increases, the direction of the velocity does not change in Fig. 1(b). The amplitude of the soliton increases at the beginning, and then, it decreases gradually. The velocity of the soliton decreases, as it comes to zero, the velocity of the soliton increases. After the velocity of the soliton comes to zero, the direction of the velocity changes in Fig. 1(c). The Fig. 2(a) shows two solitons have the overtaking interactions, and the directions of the velocities both change at the different times. The head-on interactions of two solitons are shown in Figs. 2(b) and 2(c). After the amplitudes of two solitons reach the maximum values, they quickly decrease in Fig. 2(b). From Figs. 2(b) and 2(c), the parameters µ1 (t), ν1 (t), µ2 (t) and ν2 (t) have an influence on the values and directions of the solitons. Interactions of two solitons are presented in Figs 2-4, when the parameters µ1 (t) and µ2 (t) are constants, ν1 (t) and ν2 (t) are the polynomial and trigonometric functions of t. The interactions of solitons in Fig 1-4 are determined by the time and space functions f = µ1 (t) x + ν1 (t) and h = µ2 (t) x + ν2 (t). Those are different from the types with the only time variables or the single space variables. From Solution (3), we can reveal and understand the propagation and interaction mechanisms of the solitons for Eq. (1). The soliton amplitude A, width W , velocity V and energy E R R R are, respectively, given n as A = 2−1 |e−∆ |, Wo = (c+c∗ ) e µ2 (t) dt/[(i c+ µ1 (t) e µ1 (t)dt )(−i c∗ + R R R ∗ +2 ∆ and E = q q ∗ dx. Imξ is the soliton phase, the µ1 (t) e µ1 (t)dt )], V = − φ(t)+[φ(t)] ϕ(t)+[ϕ(t)]∗ t

soliton position is d1 = −{φ(t) + [φ(t)]∗ + 2 ∆}/{ϕ(t) + [ϕ(t)]∗ } and the distance between 6

Journal Pre-proof two soliton is d12 = |d1 − d2 |.

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5. Conclusions In this paper, we have studied the inhomogeneous generalized nonlinear Schr¨odinger equation [Eq. (1)] generated by deforming the inhomogeneous Heisenberg ferromagnetic spin system through the space curve formalism and using the prolongation structure theory. The above results can be concluded as follows: 1. Via the introduction of the dependent transformation and bilinear derivative operator D, Bilinear Forms (2) and Soliton Solutions (3) for Eq. (1) have been derived. We have given the assumption of the parameter ξ for Eq. (1), which is different from those with the t-dependent coefficients, and this assumption can also be used to other x-dependent coefficients equations. Infinitely Many Conservation Laws (4) for Eq. (1) by employing the Lax pair [12] are also obtained. 2. According to Solutions (3), the propagation and interactions of the solitons have been shown in Figs. 1–4. Tnteraction mechanisms of the solitons for Eq. (1) have been presented from the soliton amplitude A, width W , velocity V and energy E. As the parameters µ1 (t), µ2 (t), ν1 (t) and ν2 (t) are chosen as the polynomial and trigonometric functions of t, the interactions of two solitons are presented in Figs. 2-4. The propagation of solitons in Fig 1-4 are determined by the time and space functions f = µ1 (t) x + ν1 (t) and h = µ2 (t) x + ν2 (t). Those are different from the types with the only time variables or the single space variables. The analytical study of Eq. (1) may help to analyze the nonlinear properties of the magnetic material and the inhomogeneous Heisenberg ferromagnetic spin chains. Acknowledgments We express our sincere thanks to the teachers and students for their helpful suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11426041.

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[16] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192-1194. [17] M. Wadati, K. Konno, Y. Ichikawa, A generalization of inverse scattering method, J. Phys. Soc. Jpn. 46 (1979) 1965-1966.

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