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Soliton molecules and novel smooth positons for the complex modified KdV equation
Journal Pre-proof Soliton molecules and novel smooth positons for the complex modified KdV equation Zhao Zhang, Xiangyu Yang, Biao Li
PII: DOI: Reference:
S0893-9659(19)30494-X https://doi.org/10.1016/j.aml.2019.106168 AML 106168
To appear in:
Applied Mathematics Letters
Received date : 22 November 2019 Revised date : 28 November 2019 Accepted date : 29 November 2019 Please cite this article as: Z. Zhang, X. Yang and B. Li, Soliton molecules and novel smooth positons for the complex modified KdV equation, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106168. 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 Published by Elsevier Ltd.
*Manuscript Click here to view linked References
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Soliton molecules and novel smooth positons for the complex modified KdV equation Zhao Zhanga,, Xiangyu Yang a,, Biao Li a,∗ a School
of Mathematics and Statistics, Ningbo University , Ningbo 315211, P. R. China
Abstract In this research, based on Darboux transformation, a molecule consisting of two identical soliton waves are firstly obtained by velocity resonance for modified KdV equation. And we also get molecules containing a plurality of solitons. Further, We study the elastic interaction between soliton molecules and typical smooth higher-order positon via semi-degenerate Darboux transformation. Last but not least, we find a new type of smooth positons called rational positons. The dynamic properties of higher-order rational positon are discussed in detail and related propositions are given in this paper. The nature of rational positons is fundamentally different from that of typical smooth positons and breather positons. The method used in this paper to get interaction solutions and rational positons can be applied to other integrable equations as well.
1. Introduction
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Keywords: Soliton molecules; Rational positon; Darboux transformation.
Recently, the problem of soliton molecules is a very hot topic. Soliton molecules, bound states of solitons, have been experimentally observed in optic systems. In 2012, soliton molecules have been numerically predicted in Bose-Einstein condensates [1]. In 2017, authors resolve the evolution of femtosecond soliton molecules in the cavity of a few-cycle mode-locked laser by means of an emerging time-stretch technique [2]. In 2018, Liu et al have experimentally observed the real-time dynamics of the entire buildup process of stable soliton molecules for the first time [3]. Very recently, Lou [4] proposes a velocity resonance mechanism to theoretically obtain soliton molecules and asymmetric solitons of (1+1)dimensional fluid systems. Just this month, Peng et al [5] have observed the temporal and spectral evolutions of the breathers in real time and breathing soliton molecules are also observed. At the same time, scholars are paying more and more attention to positon solutions. Among the various types of positons, researchers have shown an unusual fondness for smooth positons and breather positons [6–8]. Based on the degenerate Darboux transformation (DT) [9], smooth positons and breather positons can be obtained respectively by choosing seed solutions as zero and plane wave [6, 7]. However, Lou constructs a soliton molecule containing only two solitons with different heights [4]. It’s worth thinking about whether we can construct a molecule that contains two identical solitons. Can we construct a molecule that contains multiple solitons? Further, can a new type of smooth positon be constructed? This paper will take the complex modified Kortewegde Vries(mKdV) equation as an example to answer the above questions one by one. In this paper, we consider the complex mKdV equation [10] as follows:
ur
qt + q xxx + 6 |q|2 q x = 0,
(1)
Jo
which has been derived for the dynamical evolution of nonlinear lattices, plasma physics, fluid dynamics, ultrashort pulses and so on. Here, q = q(x, t) denotes a complex function of {x, t}. The Lax pairs of Eq.(1) is shown in Eq.(4) and Eq.(5) of ref. [10]. The new solution q[n] of Eq.(1) can be generated by n−fold DT from a seed solution q [10]:
N2n
∗ School
q[n] = q − 2i =
N2n , D2n
φ11 φ21 φ31 φ41 .. .
φ12 φ22 φ32 φ42 .. .
λ1 φ11 λ2 φ21 λ3 φ31 λ4 φ41 .. .
λ1 φ12 λ2 φ22 λ3 φ32 λ4 φ42 .. .
··· ··· ··· ··· .. .
φ2n1
φ2n2
λ2n φ2n1
λ2n φ2n2
···
of Mathematics and Statistics, Ningbo University, Ningbo 315211, P. R.China Email address: [email protected] ( Biao Li )
Preprint submitted to Applied Mathematics Letters
(2) λn−1 1 φ11 λn−1 2 φ21 λn−1 3 φ31 λn−1 4 φ41 .. .
λn−1 2n φ2n1
λn1 φ11 λn2 φ21 λn3 φ31 λn4 φ41 .. .
λn2n φ2n1
, November 28, 2019
Journal Pre-proof
n−1 n−1 T and D2n is given by the determinant N2n by replacing its 2n column with column vector (λn−1 1 φ12 , λ2 φ22 , · · · , λ2n φ2n2 ) , T Ψ j = [φ j1 φ j2 ] , j = 1, 2, · · · , 2n is a set of eigenfunctions with λ = λ j . And Eq.(2) satisfies the following constraints:
λ2 j = λ∗2 j−1 , φ2 j,1 = −φ∗2 j−1,2 (λ2 j−1 ), φ2 j,2 = φ∗2 j−1,1 (λ2 j−1 ), j = 1, 2, · · · , n.
(3)
In the following part of the article, based on the DT, we first obtain a molecule containing two identical soliton waves by velocity resonance. Further, molecules containing a plurality of solitons are also obtained. With some constraints, we can control the distance between these multiple solitons to be equal. Finally, we discover a new type of positon solutions called rational positons. Because these positons are rational functions. The nature of rational positons is fundamentally different from that of smooth positons and breather positons. As time goes on, the height of some crests in the rational positons gradually increases to a constant, while the rest gradually decays to the asymptotic plane. 2. Soliton molecules If the seed solution q = 0, then eigenfunctions corresponding to λ j can be provided by: φj1 Ψ j = φj2
−iλ j (4 λ j 2 t+x)−α e = iλ j (4 λ j 2 t+x)+α e
,
(4)
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where α is a real constant and λ j = ξ j + iη j . Substituting Eq.(4) into Eq.(2), then n-soliton solution can be obtained. In order to give the conditions of velocity resonance, we must give an explicit expression of one-soliton |q1−s |2 by setting λ1 = ξ1 + iη1 [7]: α |q1−s |2 = 4η21 sech2 (2η1 (−x + 4 η1 2 − 12 ξ1 2 t + )). η1
(5)
λ1 = −λ3 , λ5 = −λ7 , · · ·, λ4m−3 = −λ4m−1 , n = 2m + l.
(6)
According to Eq.(5), it is easy to know that if Eq.(2) satisfies the following resonance conditions Eq.(6), then we can obtain the interactions between l solitons and m molecules consisting of two same solitons.
It can be seen from Fig.1 (a) that the two solitons in the molecule have the same height 2 |η1 |. It is well known that the
(a)
(b)
(c)
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Figure 1: (Color online) The solution |q| to Eq.(1): (a) Soliton molecule consisting of two same solitons descried by Eq.(2) with λ1 = −λ3 = − 21 − 12 i, α = 10; (b) Elastic interaction property between two soliton molecules descried by Eq.(2) with λ1 = −λ3 = molecules consisting of four solitons descried by Eq.(2) with λ1 =
i 2 , λ3
=
√ 33 30
+
3 5 i, λ5
=
√ 15 12
+
√ 15 12
3 4 i, λ7
=
+ 43 i, λ5 = −λ7 = 1 2
√ 39 12
+ 34 i, α = 10; (c) Soliton
+ i, α = 30.
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interactions among solitons are elastic in mKdV system Eq.(1). Soliton molecules, special cases of soliton solutions Eq.(2), are also elastic in their interactions. Fig.(1) (b) shows the collision process of two soliton molecules consisting of two same solitons. There is no loss of energy after the collision, but the phase of the molecules changes. In ref. [4], the author can only obtain two solitons with the same velocity but different shapes. Here, based on the DT, we not only obtain molecules consisting of two same solitons but also molecules consisting of more than two solitons via velocity resonance. In order to obtain a molecule consisting of n solitons, the parameters in Eq.(2) is as follows: 4η22 j−1 − 12ξ22 j−1 = v0 , λ1 , λ3 · · · , λ2n−1 , j = 1, 2 · · · n,
(7)
where v0 is a real constant. Furthermore, if the distance between two adjacent solitons in the molecule is to be equal, then the constraint needs to be strengthened on the basis of Eq.(7): α α − = d0 (real constant). (8) η2 j+1 η2 j−1 2
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It can be observed from Fig.1 (c) that the four solitons in molecule are obviously different because of η1 , η3 , η5 , η7 although the velocities of solitons are the same. In addition, based on the semi-degenerate DT, we can get hybrid solutions composed of soliton molecules and nth-order smooth positons. Proposition 1: Based on the semi-degenerate DT, the interactions between molecules consisting of l solitons and mthorder smooth positons are given by N0 , qn−h = q − 2i 2n (9) D02n with 0 N2n
h i ! ! 2x−1 ∂h(i) ∂h(i) 2 , 0 , D2n = , h(x) = = (N2n )i j (λ j + ) (D2n )i j (λ j + ) 0, ∂ h(i) =0 ∂ h(i) =0 2n × 2n 2n × 2n
x≤m x > m,
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where λ1 = λ3 = · · · = λ2m−1 , n = m + l, and λ2m+1 , · · · , λ2m+2l−1 satisfies Eq.(7), [x] denotes the floor function of x and D2n , N2n are given by Eq.(2).
(a)
(b)
(c)
Figure 2: (Color online) The solution√ |q| to Eq.(1): (a) Elastic interaction between a soliton molecule and a second-order positon descried by Eq.(2) with √ 219 451 5 7 λ1 = λ3 = 2i , λ5 = 60 + 12 i, λ7 = 120 + 24 i, α = 30; (b) Elastic interaction between a soliton molecule and a third-order positon descried by Eq.(2) with
λ1 = λ3 = λ5 = 2i , λ7 =
√ 219 60
+
5 12 i, λ9
=
√ 451 120
+
7 24 i, α
= 30 ; (c) A fourth-order smooth positon descried by Eq.(2) with λ1 = λ3 = λ5 = λ7 = 2i , α = 30.
ur
Fig.2 (a)-2 (b) roughly show the elastic collision between soliton molecules and smooth positons. From them, we can observe that nothing has changed except the phase. After tedious calculation [11, 12], the trajectory of a soliton in the t− inf 8 tinf 3 3721 molecule before and after the collision is x− inf = − 8225 + 72 + 35 ln 657721 and x = − + 72 + ln respectively inf 3721 225 5 657721 in Fig.2 (a). The height of this soliton does not change after interaction with the second-order smooth positon. Before and after the collision, the trajectory equations of second-order smooth positon are x− inf = t− inf ± 21 ln 16 t−2 inf + 60 + Θ and 3721 72361 2 xinf = tinf ± 12 ln 16 tinf + 60 − Θ, Θ = 41 ln 657721 + 14 ln 5612161 , respectively. Before and after the collision, the height of the seccond-order smooth positon is 1. If l = 0 in Proposition 1, then mth-order smooth positon can be obtained via degenerate DT. Because of α , 0, the positon shown in Fig.2 (c) is significantly different from that in ref. [7, 8]. Remark 1: A more precise approximate trajectory of second-order (third-order ) smooth positon are two curves defined respectively by ln 1024 η1 6 + 9216 η1 4 ξ1 2 t2 α 2 2 (10) x = 4 η1 − 12 ξ1 t ± + , 4η1 η1 2 2 4 8 2 ln 262144 η η + 9 ξ t 1 1 1 α α (11) x = 4 η1 2 − 12 ξ1 2 t ± + , x = 4 η1 2 − 12 ξ1 2 t + , 4η1 η1 η1
Jo
rather than the Proposition 2 (Proposition 3) in ref. [7]. The height 2 |η1 | of these positon will be obtained only along the trajectory of Eq.(10) and Eq. (11) when |t| → ∞, respectively. 3. Novel smooth positons: rational positons In this section, starting with a nonzero seed solution q = c, c ∈ R, higher-order rational positons to Eq.(1) can be obtained via degenerate DT. We can get the new eigenfunctions corresponding to λ j by using the principle of superposition of the linear differential equations: q √ √ ce−i c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) + i c2 + λ 2 + iλ ei c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) j j , Ψ j = q (12) √ √ 2 −i c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) i c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) 2 i c + λ j + iλ j e + ce 3
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s = s0 + s1 2 + s2 4 + · · · + sn−1 2n−2 ,
here si , (i = 0, 1, 2, · · · , n − 1), c ∈ R are arbitrary constants. And is an infinitesimal parameter. It is trivial to find that eigenfunctions Eq.(12) are degenerate at λ2 j−1 = ic. Setting λ2 j−1 = ic + 2 , then nth-order rational positons can be obtained by higher-order Taylor expansion of q[n] Eq.(2) with respect to . Proposition 2: Based on the degenerate DT with λ2 j−1 = ic + 2 , s0 = s1 = · · · = sn−1 = 0, the nth-order rational positions qn−r are provided by N0 , qn−r = c − 2i 2n (13) D02n with 0 N2n
! ! i = 2k + 1 ∂g(i) ∂g(i) i, 0 2 2 , D2n = , g(i) = = (N2n )i j (ic + ) (D2n )i j (ic + ) g(i) g(i) ∂ =0 ∂ i − 1, i = 2k. =0 2n × 2n 2n × 2n
Here D2n , N2n , Ψ j are given by Eq.(2) and Eq.(12), and k ∈ Z. The first-order rational positon ( rational line wave) q1−r can be presented by setting n = 1 in Proposition 2: q1−r (H) = −c +
4c , H = 6 c2 t − x. 4 c2 H 2 + 1
(14)
q2−r
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It is trivial to find that |q1−r | = |c| when x → ∞, t → ∞. This means that the asymptotic plane of |q1−r | has the height |c|. The trajectory of the crest is H = 0, and the height of |q1−r | is |3c|. When n = 2, an explicit form of second-order rational positon q2−r is constructed as 4 −192 c5 6 c2 t − x − 288 c3 22 c2 t − x 6 c2 t − x + 36 c, . =c+ 64 c6 6 c2 t − x 6 − 48 c4 26 c2 t + x 6 c2 t − x 3 + 36 c2 556 c4 t2 − 68 c2 tx + 3 x2 + 9
(a)
(15)
(b)
Figure 3: (Color online) The solution |q| to Eq.(1): (a) Second-order rational positon descried by Eq.(15) with c =
√1 ; 6
(b) The 3D plot of (a).
Jo
ur
Fig.3 (a) and Fig.3 (b) show a new type of positon that is quite different from the classic one [7, 8]. First of all, this positon is a rational function; secondly, a crest in the novel positon decays to the asymptotic plane when |t| → ∞, while another gradually increases to a constant |3c|. These two properties are not found in the classic positon. The dynamic properties of rational positon are also significantly different from those of the classical smooth positon. The modulus square of a classical smooth two-positon can be decomposed as the sum of two modulus square of soliton. But this conclusion can not be true in second-order rational positon because there is an attenuated crest in it. Proposition 3: The modulus square of second-order rational positon solution for mKdV can be approximated as: 2 |q2−r |2 ≈ q1−r (H − 121/3 t1/3 ) , t → ∞, (16) 2 |q |2 ≈ q (H + 121/3 (−t)1/3 ) , t → −∞, 2−r
1−r
and two more precise approximate trajectory are defined by H − 121/3 t1/3 = 0 and H + 121/3 (−t)1/3 = 0, respectively. The proof process of Proposition 3 is roughly similar to ref. [7], so we do not provide a detailed proof of it. It is trivial to verify that the height |3c| of the second-order rational positon |q2−r | can be obtained along the trajectories of Eq.(16). Remark 2: In Proposition 3, the phase shifts of the second-order rational positon 121/3 t1/3 , 121/3 (−t)1/3 are functions of t which are different from the phase shifts of soliton solution that are usually constants. Combined with ref. [7] and Remark 1, it is found that phase shift 121/3 t1/3 of second-order rational positon is a power function of t, while phase shift of typical second-order positon is a logarithm function of t. This means that the phase shift of rational positon is more violent than that of typical positon. 4
Journal Pre-proof
In order to explain the phenomenon of Fig.3 from a deeper level, we can appropriately reduce the constraints of Proposition 2: si , 0, i = 0, 1, · · · , n − 1. In particular,if setting n = 2, s0 = 0, s1 , 0, we can get a new rational solution similar to Fig.4. As shown in Fig.4, the height of one crest in the new rational solution increases from |c| to a fixed value |3c| with time; the other crest has the opposite nature. To our surprise, the dynamics of this new rational solution with arbitrary parameter s1 are in complete agreement with Proposition 3.
(a)
(b)
(c)
Figure 4: (Color online) The evolution of a new rational solution |q| descried by Proposition 2 which reduces constraints s0 = 0, s1 = 100 with parameter selections c = √1 , n = 2: (a) t ∈ [−130, −100]; (b) t ∈ [−15, 15]; (c) t ∈ [100, 130].
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6
(a)
(b)
Figure 5: (Color online) The solution |q| to Eq.(1) : (a) Third-order rational positon descried by Eq.(13) with parameter selections c = Fourth-order rational positon descried by Eq.(13) with parameter selections c =
√1 , n 6
= 4.
√1 , n 6
= 3; (b)
As shown in Fig.5, some crests in the higher-order rational positons are also attenuated to |c| when |t| → ∞. The dynamic properties of higher-order solutions have the following propositions: Proposition 4: The modulus square of third-order rational positon solution for mKdV can be approximated as:
ur
√3 √3 2 √3 √3 2 15 t 15 t + q1−r H − 2 q − c2 , t → ∞, |q3−r |2 ≈ q1−r H + 2 q √ √ 3 3 5+3 5 3 5−5 2 2 √3 √3 √3 √3 15 −t 15 −t 2 + q H + 2 q − c2 , t → −∞ |q3−r | ≈ q1−r H − 2 q √ 1−r √ 3 3 5+3 5 3 5−5 √ 3
15 and two more precise approximate trajectory curves are defined by {H + 2 √ 3 √ √ 3 15 3 −t √ √ 3 3 5−5
t √ 5+3 5
√ 3
√ 3
15 = 0, H − 2 √ √ 3 3
t
5−5
= 0} and {H −
Jo
2
√ √ 3 15 3 −t √ √ 3 5+3 5
√ 3
(17)
= 0, H + 2
= 0}, respectively.
Proposition 5: The modulus square of fourth-order rational positon solution for mKdV can be approximated as: 2 √ √ 3 3 t 84 |q4−r |2 ≈ |q1−r (H)|2 + q1−r H − 10 q − c2 , t → ∞, r q q √ √ √ 3 3 3 9 3 21 − 7 1700 + 300 21 1700 + 300 21 − 10 2 √3 √3 84 −t |q4−r |2 ≈ |q1−r (H)|2 + q1−r H + 10 q − c2 , t → −∞ rq q √ √ 3 3 √ 3 9 3 21 − 7 1700 + 300 21 1700 + 300 21 − 10 5
(18)
Journal Pre-proof
and two more precise approximate trajectory curves are defined by {H − 10 √ 3 H = 0} and {H + 10 √ 3
3
√
21−7
√ 9
√ 3
√ 3
84 −t q √ 3√ √ 3 1700+300 21 1700+300 21−10
√ √ 3 3 t 84 q√ √ √ √ √ 3 3 9 3 21−7 1700+300 21 1700+300 21−10
= 0,
= 0, H = 0}, respectively.
It is trivial to verify that the height |3c| of the third-order rational positon |q3−r | and the fourth-order rational positon |q4−r | can all be obtained along the trajectories of Proposition 4 and 5. So Proposition 4 and 5 are correct. Remark 3: From Propositions 3, 4 and 5, we find that the phase shift of rational positon is a positive proportional function of t1/3 , (−t)1/3 , however this scale factor is very complicated. But phase shift of typical smooth positon is a function of ln(β1 tβ2 ), and β2 is getting larger as the order of the classic positon solution gets higher and higher [7]. Note that the modulus square of a nth-order rational positon can not be decomposed as the sum of n modulus square of soliton. But typical smooth positon can do this. Therefore, the dynamic properties of rational positon and typical smooth positon are significantly different. 4. Conclusion
Acknowledgment
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Although the mechanism of velocity resonance is first proposed by Lou, only molecules containing two solitons of different shapes can be obtained by bilinear method in ref. [4]. In this paper, we obtain not only molecules consisting of two same solitons but also molecules containing multiple solitons through the Darboux transformation. Although the complex mKdV equation are well known in literature [6, 10], the soliton molecules, hybrid solutions consisting of soliton molecules and typical smooth positons have not yet be found before. By specific calculation, we verify that the interaction between soliton molecules and typical smooth positons is elastic, which can be roughly obtained from Fig. 2. Based on degenerate Darboux transformation and nonzero constant seed solution, we discover a novel type of smooth positons called rational positons. The dynamic properties of rational positons have been given in detail in Propositions 3, 4 and 5. This paper expounds the essential difference between rational positon and typical smooth positon from the mathematical expressions of the solution and the dynamic properties. These differences are detailed in Remark 3. Although this method of finding rational positon is applicable to other integrable equations, some systems do not have rational psoitons, such as nonlinear Schr¨odinger equation [13]: iqt + q xx + 2 |q|2 q = 0. In addition to the bilinear method and DT, RiemannHilbert and Lie symmetry method can also be found for exact solutions [14, 15]. But it remains to be seen whether these two approaches will yield results similar to those in this paper. Meanwhile, we also hope that our results will provide some valuable information in the field of nonlinear science.
This work is supported by National Natural Science Foundation of China under Grant Nos. 11775121 and 11435005, and K.C.Wong Magna Fund in Ningbo University. The authors would like to express their sincere thanks to Professor Lou Senyue for his guidance and encouragement. Reference
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[1] K. Lakomy, R. Nath, L. Santos, Spontaneous crystallization and filamentation of solitons in dipolar condensates, Phys. Rev. A 85 (2012) 033618 [2] G. Herink, F. Kurtz, B. Jalali, D. R. Solli, C. Ropers, Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules, Science 356 (2017) 50-54 [3] X.M. Liu, X.K. Yao, Y.D. Cui, Real-Time Observation of the Buildup of Soliton Molecules, Phys. Rev. Lett. 121 (2018) 023905 [4] S.Y. Lou, Soliton molecules and asymmetric solitons in fluid systems via velocity resonance, (2019) arXiv:1909.03399 [nlin .SI] [5] J. Peng, S. Boscolo, Z. Zhao, H. Zeng, Breathing dissipative solitons inmode-locked fiber lasers, Sci. Adv. 5 (2019) eaax1110 [6] D.Q. Qiu, W.G. Cheng, The nth-order degenrate breather solution for the Kundu-Eckhaus equation, Appl. Math. Lett. 98 (2019) 13-21 [7] W. Liu, Y.S. Zhang, J.S. He, Dynamics of the smooth positons of the complex modified KdV equation, Wave Random Complex 28 (2018) 203-214 [8] W.J. Song, S.W. Xu, M.H. Li, J.S. He, Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schr¨odinger equation, Nonlinear Dyn. 97 (2019) 2135-2145 [9] W.Q. Peng, S.T. Tian, T.T. Zhang, Breather waves, high-order rogue waves and their dynamics in the coupled nonlinear Schr¨odinger equations with alternate signs of nonlinearities, EPL-Europhys Lett. 127 (2019) 50005 [10] J.S. He, L.H. Wang, L.J. Li, K. Porsezian, R. Erd´elyi, Few-cycle optical rogue waves: Complex modified Korteweg-de Vries equation, Phys. Rev. E 89 (2014) 062917 [11] Z. Zhang, X.Y. Yang, W.T. Li, B. Li, Trajectory equation of a lump before and after collision with line, lump, and breather waves for (2+1)-dimensional Kadomtsev-Petviashvili equation, Chin. Phys. B 28 (2019) 110201 [12] Z. Zhang, S.X. Yang, B. Li, Soliton Molecules, Asymmetric Solitons and Hybrid Solutions for (2+1)-Dimensional Fifth-Order KdV Equation, Chin. Phys. Lett. 36 (2019) 120501 [13] J.S. He, H.R. Zhang, L.H. Wang, K. Porsezian, A.S. Fokas, Generating mechanism for higher-order rogue waves, Phys. Rev. E 87 (2013) 052914 [14] S.F. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation, Appl. Math. Lett. 100 (2020) 106056 [15] W.Q. Peng, S.F. Tian, X.B. Wang, T.T. Zhang, Y. Fang, Riemann-Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schr¨odinger equations, J. Geom. Phys. 146 (2019) 103508
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*Author Contributions Section
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Author Contributions Section
Jo
ur
na lP repr oo f
1. The author Zhao Zhang is mainly responsible for the calculation part of the paper. 2. The author Xiangyu Yang is mainly responsible for searching the literature and verifying the calculation results. 3. The author Biao Li is mainly responsible for the theoretical method guidance and thesis writing.
PII: DOI: Reference:
S0893-9659(19)30494-X https://doi.org/10.1016/j.aml.2019.106168 AML 106168
To appear in:
Applied Mathematics Letters
Received date : 22 November 2019 Revised date : 28 November 2019 Accepted date : 29 November 2019 Please cite this article as: Z. Zhang, X. Yang and B. Li, Soliton molecules and novel smooth positons for the complex modified KdV equation, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106168. 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 Published by Elsevier Ltd.
*Manuscript Click here to view linked References
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Soliton molecules and novel smooth positons for the complex modified KdV equation Zhao Zhanga,, Xiangyu Yang a,, Biao Li a,∗ a School
of Mathematics and Statistics, Ningbo University , Ningbo 315211, P. R. China
Abstract In this research, based on Darboux transformation, a molecule consisting of two identical soliton waves are firstly obtained by velocity resonance for modified KdV equation. And we also get molecules containing a plurality of solitons. Further, We study the elastic interaction between soliton molecules and typical smooth higher-order positon via semi-degenerate Darboux transformation. Last but not least, we find a new type of smooth positons called rational positons. The dynamic properties of higher-order rational positon are discussed in detail and related propositions are given in this paper. The nature of rational positons is fundamentally different from that of typical smooth positons and breather positons. The method used in this paper to get interaction solutions and rational positons can be applied to other integrable equations as well.
1. Introduction
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Keywords: Soliton molecules; Rational positon; Darboux transformation.
Recently, the problem of soliton molecules is a very hot topic. Soliton molecules, bound states of solitons, have been experimentally observed in optic systems. In 2012, soliton molecules have been numerically predicted in Bose-Einstein condensates [1]. In 2017, authors resolve the evolution of femtosecond soliton molecules in the cavity of a few-cycle mode-locked laser by means of an emerging time-stretch technique [2]. In 2018, Liu et al have experimentally observed the real-time dynamics of the entire buildup process of stable soliton molecules for the first time [3]. Very recently, Lou [4] proposes a velocity resonance mechanism to theoretically obtain soliton molecules and asymmetric solitons of (1+1)dimensional fluid systems. Just this month, Peng et al [5] have observed the temporal and spectral evolutions of the breathers in real time and breathing soliton molecules are also observed. At the same time, scholars are paying more and more attention to positon solutions. Among the various types of positons, researchers have shown an unusual fondness for smooth positons and breather positons [6–8]. Based on the degenerate Darboux transformation (DT) [9], smooth positons and breather positons can be obtained respectively by choosing seed solutions as zero and plane wave [6, 7]. However, Lou constructs a soliton molecule containing only two solitons with different heights [4]. It’s worth thinking about whether we can construct a molecule that contains two identical solitons. Can we construct a molecule that contains multiple solitons? Further, can a new type of smooth positon be constructed? This paper will take the complex modified Kortewegde Vries(mKdV) equation as an example to answer the above questions one by one. In this paper, we consider the complex mKdV equation [10] as follows:
ur
qt + q xxx + 6 |q|2 q x = 0,
(1)
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which has been derived for the dynamical evolution of nonlinear lattices, plasma physics, fluid dynamics, ultrashort pulses and so on. Here, q = q(x, t) denotes a complex function of {x, t}. The Lax pairs of Eq.(1) is shown in Eq.(4) and Eq.(5) of ref. [10]. The new solution q[n] of Eq.(1) can be generated by n−fold DT from a seed solution q [10]:
N2n
∗ School
q[n] = q − 2i =
N2n , D2n
φ11 φ21 φ31 φ41 .. .
φ12 φ22 φ32 φ42 .. .
λ1 φ11 λ2 φ21 λ3 φ31 λ4 φ41 .. .
λ1 φ12 λ2 φ22 λ3 φ32 λ4 φ42 .. .
··· ··· ··· ··· .. .
φ2n1
φ2n2
λ2n φ2n1
λ2n φ2n2
···
of Mathematics and Statistics, Ningbo University, Ningbo 315211, P. R.China Email address: [email protected] ( Biao Li )
Preprint submitted to Applied Mathematics Letters
(2) λn−1 1 φ11 λn−1 2 φ21 λn−1 3 φ31 λn−1 4 φ41 .. .
λn−1 2n φ2n1
λn1 φ11 λn2 φ21 λn3 φ31 λn4 φ41 .. .
λn2n φ2n1
, November 28, 2019
Journal Pre-proof
n−1 n−1 T and D2n is given by the determinant N2n by replacing its 2n column with column vector (λn−1 1 φ12 , λ2 φ22 , · · · , λ2n φ2n2 ) , T Ψ j = [φ j1 φ j2 ] , j = 1, 2, · · · , 2n is a set of eigenfunctions with λ = λ j . And Eq.(2) satisfies the following constraints:
λ2 j = λ∗2 j−1 , φ2 j,1 = −φ∗2 j−1,2 (λ2 j−1 ), φ2 j,2 = φ∗2 j−1,1 (λ2 j−1 ), j = 1, 2, · · · , n.
(3)
In the following part of the article, based on the DT, we first obtain a molecule containing two identical soliton waves by velocity resonance. Further, molecules containing a plurality of solitons are also obtained. With some constraints, we can control the distance between these multiple solitons to be equal. Finally, we discover a new type of positon solutions called rational positons. Because these positons are rational functions. The nature of rational positons is fundamentally different from that of smooth positons and breather positons. As time goes on, the height of some crests in the rational positons gradually increases to a constant, while the rest gradually decays to the asymptotic plane. 2. Soliton molecules If the seed solution q = 0, then eigenfunctions corresponding to λ j can be provided by: φj1 Ψ j = φj2
−iλ j (4 λ j 2 t+x)−α e = iλ j (4 λ j 2 t+x)+α e
,
(4)
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where α is a real constant and λ j = ξ j + iη j . Substituting Eq.(4) into Eq.(2), then n-soliton solution can be obtained. In order to give the conditions of velocity resonance, we must give an explicit expression of one-soliton |q1−s |2 by setting λ1 = ξ1 + iη1 [7]: α |q1−s |2 = 4η21 sech2 (2η1 (−x + 4 η1 2 − 12 ξ1 2 t + )). η1
(5)
λ1 = −λ3 , λ5 = −λ7 , · · ·, λ4m−3 = −λ4m−1 , n = 2m + l.
(6)
According to Eq.(5), it is easy to know that if Eq.(2) satisfies the following resonance conditions Eq.(6), then we can obtain the interactions between l solitons and m molecules consisting of two same solitons.
It can be seen from Fig.1 (a) that the two solitons in the molecule have the same height 2 |η1 |. It is well known that the
(a)
(b)
(c)
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Figure 1: (Color online) The solution |q| to Eq.(1): (a) Soliton molecule consisting of two same solitons descried by Eq.(2) with λ1 = −λ3 = − 21 − 12 i, α = 10; (b) Elastic interaction property between two soliton molecules descried by Eq.(2) with λ1 = −λ3 = molecules consisting of four solitons descried by Eq.(2) with λ1 =
i 2 , λ3
=
√ 33 30
+
3 5 i, λ5
=
√ 15 12
+
√ 15 12
3 4 i, λ7
=
+ 43 i, λ5 = −λ7 = 1 2
√ 39 12
+ 34 i, α = 10; (c) Soliton
+ i, α = 30.
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interactions among solitons are elastic in mKdV system Eq.(1). Soliton molecules, special cases of soliton solutions Eq.(2), are also elastic in their interactions. Fig.(1) (b) shows the collision process of two soliton molecules consisting of two same solitons. There is no loss of energy after the collision, but the phase of the molecules changes. In ref. [4], the author can only obtain two solitons with the same velocity but different shapes. Here, based on the DT, we not only obtain molecules consisting of two same solitons but also molecules consisting of more than two solitons via velocity resonance. In order to obtain a molecule consisting of n solitons, the parameters in Eq.(2) is as follows: 4η22 j−1 − 12ξ22 j−1 = v0 , λ1 , λ3 · · · , λ2n−1 , j = 1, 2 · · · n,
(7)
where v0 is a real constant. Furthermore, if the distance between two adjacent solitons in the molecule is to be equal, then the constraint needs to be strengthened on the basis of Eq.(7): α α − = d0 (real constant). (8) η2 j+1 η2 j−1 2
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It can be observed from Fig.1 (c) that the four solitons in molecule are obviously different because of η1 , η3 , η5 , η7 although the velocities of solitons are the same. In addition, based on the semi-degenerate DT, we can get hybrid solutions composed of soliton molecules and nth-order smooth positons. Proposition 1: Based on the semi-degenerate DT, the interactions between molecules consisting of l solitons and mthorder smooth positons are given by N0 , qn−h = q − 2i 2n (9) D02n with 0 N2n
h i ! ! 2x−1 ∂h(i) ∂h(i) 2 , 0 , D2n = , h(x) = = (N2n )i j (λ j + ) (D2n )i j (λ j + ) 0, ∂ h(i) =0 ∂ h(i) =0 2n × 2n 2n × 2n
x≤m x > m,
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where λ1 = λ3 = · · · = λ2m−1 , n = m + l, and λ2m+1 , · · · , λ2m+2l−1 satisfies Eq.(7), [x] denotes the floor function of x and D2n , N2n are given by Eq.(2).
(a)
(b)
(c)
Figure 2: (Color online) The solution√ |q| to Eq.(1): (a) Elastic interaction between a soliton molecule and a second-order positon descried by Eq.(2) with √ 219 451 5 7 λ1 = λ3 = 2i , λ5 = 60 + 12 i, λ7 = 120 + 24 i, α = 30; (b) Elastic interaction between a soliton molecule and a third-order positon descried by Eq.(2) with
λ1 = λ3 = λ5 = 2i , λ7 =
√ 219 60
+
5 12 i, λ9
=
√ 451 120
+
7 24 i, α
= 30 ; (c) A fourth-order smooth positon descried by Eq.(2) with λ1 = λ3 = λ5 = λ7 = 2i , α = 30.
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Fig.2 (a)-2 (b) roughly show the elastic collision between soliton molecules and smooth positons. From them, we can observe that nothing has changed except the phase. After tedious calculation [11, 12], the trajectory of a soliton in the t− inf 8 tinf 3 3721 molecule before and after the collision is x− inf = − 8225 + 72 + 35 ln 657721 and x = − + 72 + ln respectively inf 3721 225 5 657721 in Fig.2 (a). The height of this soliton does not change after interaction with the second-order smooth positon. Before and after the collision, the trajectory equations of second-order smooth positon are x− inf = t− inf ± 21 ln 16 t−2 inf + 60 + Θ and 3721 72361 2 xinf = tinf ± 12 ln 16 tinf + 60 − Θ, Θ = 41 ln 657721 + 14 ln 5612161 , respectively. Before and after the collision, the height of the seccond-order smooth positon is 1. If l = 0 in Proposition 1, then mth-order smooth positon can be obtained via degenerate DT. Because of α , 0, the positon shown in Fig.2 (c) is significantly different from that in ref. [7, 8]. Remark 1: A more precise approximate trajectory of second-order (third-order ) smooth positon are two curves defined respectively by ln 1024 η1 6 + 9216 η1 4 ξ1 2 t2 α 2 2 (10) x = 4 η1 − 12 ξ1 t ± + , 4η1 η1 2 2 4 8 2 ln 262144 η η + 9 ξ t 1 1 1 α α (11) x = 4 η1 2 − 12 ξ1 2 t ± + , x = 4 η1 2 − 12 ξ1 2 t + , 4η1 η1 η1
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rather than the Proposition 2 (Proposition 3) in ref. [7]. The height 2 |η1 | of these positon will be obtained only along the trajectory of Eq.(10) and Eq. (11) when |t| → ∞, respectively. 3. Novel smooth positons: rational positons In this section, starting with a nonzero seed solution q = c, c ∈ R, higher-order rational positons to Eq.(1) can be obtained via degenerate DT. We can get the new eigenfunctions corresponding to λ j by using the principle of superposition of the linear differential equations: q √ √ ce−i c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) + i c2 + λ 2 + iλ ei c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) j j , Ψ j = q (12) √ √ 2 −i c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) i c2 +λ j 2 (2 c2 t−4 tλ j 2 −s−x) 2 i c + λ j + iλ j e + ce 3
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s = s0 + s1 2 + s2 4 + · · · + sn−1 2n−2 ,
here si , (i = 0, 1, 2, · · · , n − 1), c ∈ R are arbitrary constants. And is an infinitesimal parameter. It is trivial to find that eigenfunctions Eq.(12) are degenerate at λ2 j−1 = ic. Setting λ2 j−1 = ic + 2 , then nth-order rational positons can be obtained by higher-order Taylor expansion of q[n] Eq.(2) with respect to . Proposition 2: Based on the degenerate DT with λ2 j−1 = ic + 2 , s0 = s1 = · · · = sn−1 = 0, the nth-order rational positions qn−r are provided by N0 , qn−r = c − 2i 2n (13) D02n with 0 N2n
! ! i = 2k + 1 ∂g(i) ∂g(i) i, 0 2 2 , D2n = , g(i) = = (N2n )i j (ic + ) (D2n )i j (ic + ) g(i) g(i) ∂ =0 ∂ i − 1, i = 2k. =0 2n × 2n 2n × 2n
Here D2n , N2n , Ψ j are given by Eq.(2) and Eq.(12), and k ∈ Z. The first-order rational positon ( rational line wave) q1−r can be presented by setting n = 1 in Proposition 2: q1−r (H) = −c +
4c , H = 6 c2 t − x. 4 c2 H 2 + 1
(14)
q2−r
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It is trivial to find that |q1−r | = |c| when x → ∞, t → ∞. This means that the asymptotic plane of |q1−r | has the height |c|. The trajectory of the crest is H = 0, and the height of |q1−r | is |3c|. When n = 2, an explicit form of second-order rational positon q2−r is constructed as 4 −192 c5 6 c2 t − x − 288 c3 22 c2 t − x 6 c2 t − x + 36 c, . =c+ 64 c6 6 c2 t − x 6 − 48 c4 26 c2 t + x 6 c2 t − x 3 + 36 c2 556 c4 t2 − 68 c2 tx + 3 x2 + 9
(a)
(15)
(b)
Figure 3: (Color online) The solution |q| to Eq.(1): (a) Second-order rational positon descried by Eq.(15) with c =
√1 ; 6
(b) The 3D plot of (a).
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Fig.3 (a) and Fig.3 (b) show a new type of positon that is quite different from the classic one [7, 8]. First of all, this positon is a rational function; secondly, a crest in the novel positon decays to the asymptotic plane when |t| → ∞, while another gradually increases to a constant |3c|. These two properties are not found in the classic positon. The dynamic properties of rational positon are also significantly different from those of the classical smooth positon. The modulus square of a classical smooth two-positon can be decomposed as the sum of two modulus square of soliton. But this conclusion can not be true in second-order rational positon because there is an attenuated crest in it. Proposition 3: The modulus square of second-order rational positon solution for mKdV can be approximated as: 2 |q2−r |2 ≈ q1−r (H − 121/3 t1/3 ) , t → ∞, (16) 2 |q |2 ≈ q (H + 121/3 (−t)1/3 ) , t → −∞, 2−r
1−r
and two more precise approximate trajectory are defined by H − 121/3 t1/3 = 0 and H + 121/3 (−t)1/3 = 0, respectively. The proof process of Proposition 3 is roughly similar to ref. [7], so we do not provide a detailed proof of it. It is trivial to verify that the height |3c| of the second-order rational positon |q2−r | can be obtained along the trajectories of Eq.(16). Remark 2: In Proposition 3, the phase shifts of the second-order rational positon 121/3 t1/3 , 121/3 (−t)1/3 are functions of t which are different from the phase shifts of soliton solution that are usually constants. Combined with ref. [7] and Remark 1, it is found that phase shift 121/3 t1/3 of second-order rational positon is a power function of t, while phase shift of typical second-order positon is a logarithm function of t. This means that the phase shift of rational positon is more violent than that of typical positon. 4
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In order to explain the phenomenon of Fig.3 from a deeper level, we can appropriately reduce the constraints of Proposition 2: si , 0, i = 0, 1, · · · , n − 1. In particular,if setting n = 2, s0 = 0, s1 , 0, we can get a new rational solution similar to Fig.4. As shown in Fig.4, the height of one crest in the new rational solution increases from |c| to a fixed value |3c| with time; the other crest has the opposite nature. To our surprise, the dynamics of this new rational solution with arbitrary parameter s1 are in complete agreement with Proposition 3.
(a)
(b)
(c)
Figure 4: (Color online) The evolution of a new rational solution |q| descried by Proposition 2 which reduces constraints s0 = 0, s1 = 100 with parameter selections c = √1 , n = 2: (a) t ∈ [−130, −100]; (b) t ∈ [−15, 15]; (c) t ∈ [100, 130].
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6
(a)
(b)
Figure 5: (Color online) The solution |q| to Eq.(1) : (a) Third-order rational positon descried by Eq.(13) with parameter selections c = Fourth-order rational positon descried by Eq.(13) with parameter selections c =
√1 , n 6
= 4.
√1 , n 6
= 3; (b)
As shown in Fig.5, some crests in the higher-order rational positons are also attenuated to |c| when |t| → ∞. The dynamic properties of higher-order solutions have the following propositions: Proposition 4: The modulus square of third-order rational positon solution for mKdV can be approximated as:
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√3 √3 2 √3 √3 2 15 t 15 t + q1−r H − 2 q − c2 , t → ∞, |q3−r |2 ≈ q1−r H + 2 q √ √ 3 3 5+3 5 3 5−5 2 2 √3 √3 √3 √3 15 −t 15 −t 2 + q H + 2 q − c2 , t → −∞ |q3−r | ≈ q1−r H − 2 q √ 1−r √ 3 3 5+3 5 3 5−5 √ 3
15 and two more precise approximate trajectory curves are defined by {H + 2 √ 3 √ √ 3 15 3 −t √ √ 3 3 5−5
t √ 5+3 5
√ 3
√ 3
15 = 0, H − 2 √ √ 3 3
t
5−5
= 0} and {H −
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2
√ √ 3 15 3 −t √ √ 3 5+3 5
√ 3
(17)
= 0, H + 2
= 0}, respectively.
Proposition 5: The modulus square of fourth-order rational positon solution for mKdV can be approximated as: 2 √ √ 3 3 t 84 |q4−r |2 ≈ |q1−r (H)|2 + q1−r H − 10 q − c2 , t → ∞, r q q √ √ √ 3 3 3 9 3 21 − 7 1700 + 300 21 1700 + 300 21 − 10 2 √3 √3 84 −t |q4−r |2 ≈ |q1−r (H)|2 + q1−r H + 10 q − c2 , t → −∞ rq q √ √ 3 3 √ 3 9 3 21 − 7 1700 + 300 21 1700 + 300 21 − 10 5
(18)
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and two more precise approximate trajectory curves are defined by {H − 10 √ 3 H = 0} and {H + 10 √ 3
3
√
21−7
√ 9
√ 3
√ 3
84 −t q √ 3√ √ 3 1700+300 21 1700+300 21−10
√ √ 3 3 t 84 q√ √ √ √ √ 3 3 9 3 21−7 1700+300 21 1700+300 21−10
= 0,
= 0, H = 0}, respectively.
It is trivial to verify that the height |3c| of the third-order rational positon |q3−r | and the fourth-order rational positon |q4−r | can all be obtained along the trajectories of Proposition 4 and 5. So Proposition 4 and 5 are correct. Remark 3: From Propositions 3, 4 and 5, we find that the phase shift of rational positon is a positive proportional function of t1/3 , (−t)1/3 , however this scale factor is very complicated. But phase shift of typical smooth positon is a function of ln(β1 tβ2 ), and β2 is getting larger as the order of the classic positon solution gets higher and higher [7]. Note that the modulus square of a nth-order rational positon can not be decomposed as the sum of n modulus square of soliton. But typical smooth positon can do this. Therefore, the dynamic properties of rational positon and typical smooth positon are significantly different. 4. Conclusion
Acknowledgment
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Although the mechanism of velocity resonance is first proposed by Lou, only molecules containing two solitons of different shapes can be obtained by bilinear method in ref. [4]. In this paper, we obtain not only molecules consisting of two same solitons but also molecules containing multiple solitons through the Darboux transformation. Although the complex mKdV equation are well known in literature [6, 10], the soliton molecules, hybrid solutions consisting of soliton molecules and typical smooth positons have not yet be found before. By specific calculation, we verify that the interaction between soliton molecules and typical smooth positons is elastic, which can be roughly obtained from Fig. 2. Based on degenerate Darboux transformation and nonzero constant seed solution, we discover a novel type of smooth positons called rational positons. The dynamic properties of rational positons have been given in detail in Propositions 3, 4 and 5. This paper expounds the essential difference between rational positon and typical smooth positon from the mathematical expressions of the solution and the dynamic properties. These differences are detailed in Remark 3. Although this method of finding rational positon is applicable to other integrable equations, some systems do not have rational psoitons, such as nonlinear Schr¨odinger equation [13]: iqt + q xx + 2 |q|2 q = 0. In addition to the bilinear method and DT, RiemannHilbert and Lie symmetry method can also be found for exact solutions [14, 15]. But it remains to be seen whether these two approaches will yield results similar to those in this paper. Meanwhile, we also hope that our results will provide some valuable information in the field of nonlinear science.
This work is supported by National Natural Science Foundation of China under Grant Nos. 11775121 and 11435005, and K.C.Wong Magna Fund in Ningbo University. The authors would like to express their sincere thanks to Professor Lou Senyue for his guidance and encouragement. Reference
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[1] K. Lakomy, R. Nath, L. Santos, Spontaneous crystallization and filamentation of solitons in dipolar condensates, Phys. Rev. A 85 (2012) 033618 [2] G. Herink, F. Kurtz, B. Jalali, D. R. Solli, C. Ropers, Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules, Science 356 (2017) 50-54 [3] X.M. Liu, X.K. Yao, Y.D. Cui, Real-Time Observation of the Buildup of Soliton Molecules, Phys. Rev. Lett. 121 (2018) 023905 [4] S.Y. Lou, Soliton molecules and asymmetric solitons in fluid systems via velocity resonance, (2019) arXiv:1909.03399 [nlin .SI] [5] J. Peng, S. Boscolo, Z. Zhao, H. Zeng, Breathing dissipative solitons inmode-locked fiber lasers, Sci. Adv. 5 (2019) eaax1110 [6] D.Q. Qiu, W.G. Cheng, The nth-order degenrate breather solution for the Kundu-Eckhaus equation, Appl. Math. Lett. 98 (2019) 13-21 [7] W. Liu, Y.S. Zhang, J.S. He, Dynamics of the smooth positons of the complex modified KdV equation, Wave Random Complex 28 (2018) 203-214 [8] W.J. Song, S.W. Xu, M.H. Li, J.S. He, Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schr¨odinger equation, Nonlinear Dyn. 97 (2019) 2135-2145 [9] W.Q. Peng, S.T. Tian, T.T. Zhang, Breather waves, high-order rogue waves and their dynamics in the coupled nonlinear Schr¨odinger equations with alternate signs of nonlinearities, EPL-Europhys Lett. 127 (2019) 50005 [10] J.S. He, L.H. Wang, L.J. Li, K. Porsezian, R. Erd´elyi, Few-cycle optical rogue waves: Complex modified Korteweg-de Vries equation, Phys. Rev. E 89 (2014) 062917 [11] Z. Zhang, X.Y. Yang, W.T. Li, B. Li, Trajectory equation of a lump before and after collision with line, lump, and breather waves for (2+1)-dimensional Kadomtsev-Petviashvili equation, Chin. Phys. B 28 (2019) 110201 [12] Z. Zhang, S.X. Yang, B. Li, Soliton Molecules, Asymmetric Solitons and Hybrid Solutions for (2+1)-Dimensional Fifth-Order KdV Equation, Chin. Phys. Lett. 36 (2019) 120501 [13] J.S. He, H.R. Zhang, L.H. Wang, K. Porsezian, A.S. Fokas, Generating mechanism for higher-order rogue waves, Phys. Rev. E 87 (2013) 052914 [14] S.F. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation, Appl. Math. Lett. 100 (2020) 106056 [15] W.Q. Peng, S.F. Tian, X.B. Wang, T.T. Zhang, Y. Fang, Riemann-Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schr¨odinger equations, J. Geom. Phys. 146 (2019) 103508
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*Author Contributions Section
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Author Contributions Section
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1. The author Zhao Zhang is mainly responsible for the calculation part of the paper. 2. The author Xiangyu Yang is mainly responsible for searching the literature and verifying the calculation results. 3. The author Biao Li is mainly responsible for the theoretical method guidance and thesis writing.