Nonlinear dynamics for different nonautonomous wave structure solutions

Kun-Qiong Li

doi:10.1515/phys-2022-0050

Open Access Published by De Gruyter Open Access June 18, 2022

Nonlinear dynamics for different nonautonomous wave structure solutions

Kun-Qiong Li

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0050

Abstract

Based on the positive quadratic function method, the rich nonautonomous solutions of a generalized (2+1)-dimensional variable-coefficient breaking soliton equation with different wave structures are given. In this case, due to the influence of nonlinearity and dispersion, the characteristics, amplitude and velocity of nonautonomous wave will change with time. The breather wave and the interaction among lump wave, solitary wave and periodic wave solutions are studied. For different choices of arbitrary functions in these solutions, the corresponding dynamic properties are demonstrated.

Keywords: positive quadratic function method; lump wave; breather wave; breaking soliton equation

1 Introduction

In recent years, with the development of symbolic computation, people have begun to pay attention to the relevant theories of lump wave solutions [1,2, 3,4]. In 2015, Ma [5] proposed a method of directly using the Hirota bilinear method to solve lump wave solutions and gave theoretical proofs and derivations, pushing the research of lump wave solutions to a new stage. At present, many researchers have successfully constructed lump wave solutions and interaction solutions of multiple high-dimensional nonlinear development equations using this method [6,7, 8,9]. The research of these solutions has important significance and prospects for many high-dimensional nonlinear problems in mathematics, physics and other fields.

Variable coefficient integrability systems can more clearly describe real-world phenomena, such as in the context of ocean waves, the temporal variability of the variable coefficient may be due to the pressure dependence of the coefficient of thermal expansion of seawater, coupled with large-scale forward changes in ocean temperature–salinity relationship and other dynamic conditions [10]. In this article, under investigation is the generalized (2+1)-dimensional variable-coefficient breaking soliton equation [11]

(1) τ 1 ( t ) u x u x x + τ 4 ( t ) ( u x u x y + u y u x x ) + τ 3 ( t ) u x x x y + τ 2 ( t ) u x x x x + u x t = 0 ,

where u = u ( x , y , t ) represents the interactions among two Riemann waves propagating along the y - or z -axis and a long wave propagating along the x -axis. τ 1 ( t ) , τ 2 ( t ) , τ 3 ( t ) and τ 4 ( t ) are arbitrary functions. Osman [12] presented the multi-soliton solutions of Eq. (1) by the generalized unified method. Li et al. [11] obtained the breather wave solutions and lump solutions of Eq. (1). However, the interaction among lump wave, solitary wave and periodic wave solutions has not been investigated, which will become our main work.

The article is organized as follows. Section 2 investigates the interaction between lump wave and solitary wave; Section 3 studies the interaction between lump wave and periodic wave; Section 4 obtains the breather wave solutions; Section 5 gives a conclusion.

2 Interaction between lump wave and solitary wave

Under the transformation

(2) τ 1 ( t ) = 6 τ 2 ( t ) , τ 4 ( t ) = 3 τ 3 ( t ) , u = 2 ( ln κ ) x , κ = κ ( x , y , t ) .

Eq. (2) becomes

(3) [ τ 2 ( t ) D x 4 + τ 3 ( t ) D x 3 D y + D t D x ] κ ⋅ κ = 3 τ 2 ( t ) κ x x 2 + 3 τ 3 ( t ) κ x y κ x x − 3 τ 3 ( t ) κ x κ x x y − τ 3 ( t ) κ y κ x x x − 4 τ 2 ( t ) κ x κ x x x + κ ( τ 3 ( t ) κ x x x y + τ 2 ( t ) κ x x x x + κ x t ) − κ t κ x = 0 .

In order to investigate the interaction between lump wave and solitary wave of Eq. (1), the following assumptions are usually made

(4) κ = ν 7 ( t ) + ∫ ν 3 ( t ) d t + ν 1 x + ν 2 y 2 + ∫ ν 6 ( t ) d t + ν 4 x + ν 5 y 2 + ϕ 1 ( t ) e ∫ φ 3 ( t ) d t + φ 1 x + φ 2 y + ϕ 2 ( t ) e − ∫ φ 3 ( t ) d t − φ 1 x − φ 2 y ,

where ν i ( i = 1 , 2 , 4 , 5 ) and φ i ( i = 1 , 2 ) are unknown constants. ν i ( t ) ( i = 3 , 6 , 7 ) , φ 3 ( t ) and ϕ i ( t ) ( i = 1 , 2 ) are unknown functions. Substituting Eq. (4) into Eq. (3), we obtain

(5) ν 5 = ν 2 ν 4 ν 1 , ν 6 ( t ) = − ν 1 ν 3 ( t ) ν 4 , φ 2 = ν 2 φ 1 ν 1 , τ 2 ( t ) = − ν 2 τ 3 ( t ) ν 1 , ϕ 1 ( t ) = η 1 e − ∫ φ 3 ( t ) d t , ϕ 2 ( t ) = η 2 e ∫ φ 3 ( t ) d t , ν 7 ( t ) = ∫ − 2 ( ν 1 2 + ν 4 2 ) ν 3 ( t ) ∫ ν 3 ( t ) d t ν 4 2 d t ,

where η i ( i = 1 , 2 ) is the integral constant. Substituting Eqs. (4) and (5) into Eq. (2), the interaction solution between lump wave and solitary wave of Eq. (1) is expressed as

(6) u = 2 2 ν 1 ∫ ν 3 ( t ) d t + ν 1 x + ν 2 y + 2 ν 4 ν 4 x − ν 1 ∫ ν 3 ( t ) d t ν 4 + ν 2 ν 4 y ν 1 + η 1 φ 1 e φ 1 x + ν 2 φ 1 y ν 1 − η 2 φ 1 e − φ 1 x − ν 2 φ 1 y ν 1 ∫ ν 3 ( t ) d t + ν 1 x + ν 2 y 2 + ν 4 x − ν 1 ∫ ν 3 ( t ) d t ν 4 + ν 2 ν 4 y ν 1 2 + η 1 e φ 1 x + ν 2 φ 1 y ν 1 + η 2 e − φ 1 x − ν 2 φ 1 y ν 1 − ∫ 2 ( ν 1 2 + ν 4 2 ) ν 3 ( t ) ∫ ν 3 ( t ) d t ν 4 2 d t .

When η 1 = η 2 = 0 , Eq. (6) describes a lump wave, which has been studied in ref. [11]. When η 1 = 0 , η 2 ≠ 0 , Eq. (6) represents the interaction solution between lump wave and solitary wave (Figure 1), which has not been discussed in other literature. When η 1 ≠ 0 and η 2 ≠ 0 , Eq. (6) means the interaction solution between lump wave and two solitary waves (Figure 2), which has not been studied in other literature.

$Figure 1 Solution (6) with ν 1 = 2 {\nu }_{1}=2 , ν 2 = ν 4 = ν 3 ( t ) = φ 3 ( t ) = 1 {\nu }_{2}={\nu }_{4}={\nu }_{3}\left(t)={\varphi }_{3}\left(t)=1 , φ 1 = η 1 = 3 {\varphi }_{1}={\eta }_{1}=3 , η 2 = 0 {\eta }_{2}=0 ; (a) t = − 2 t=-2 , (b) t = 0 t=0 , and (c) t = 2 t=2 .$

Figure 1

Solution (6) with ν 1 = 2 , ν 2 = ν 4 = ν 3 ( t ) = φ 3 ( t ) = 1 , φ 1 = η 1 = 3 , η 2 = 0 ; (a) t = − 2 , (b) t = 0 , and (c) t = 2 .

$Figure 2 Solution (6) with ν 2 = ν 4 = ν 3 ( t ) = φ 3 ( t ) = 1 {\nu }_{2}={\nu }_{4}={\nu }_{3}\left(t)={\varphi }_{3}\left(t)=1 , φ 1 = η 1 = η 2 = 3 {\varphi }_{1}={\eta }_{1}={\eta }_{2}=3 , ν 1 = 2 {\nu }_{1}=2 ; (a) t = − 5 t=-5 , (b) t = 0 t=0 , and (c) t = 5 t=5 .$

Figure 2

Solution (6) with ν 2 = ν 4 = ν 3 ( t ) = φ 3 ( t ) = 1 , φ 1 = η 1 = η 2 = 3 , ν 1 = 2 ; (a) t = − 5 , (b) t = 0 , and (c) t = 5 .

3 Interaction between lump wave and periodic wave

In order to study the interaction between lump wave and periodic wave of Eq. (1), the following assumptions are usually made

(7) κ = ν 7 ( t ) + ∫ ν 3 ( t ) d t + ν 1 x + ν 2 y 2 + ∫ ν 6 ( t ) d t + ν 4 x + ν 5 y 2 + ϕ 1 ( t ) cos ∫ φ 3 ( t ) d t + φ 1 x + φ 2 y .

Substituting Eq. (7) into Eq. (3), we obtain

(8) ν 5 = ν 2 ν 4 ν 1 , ν 6 ( t ) = − ν 1 ν 3 ( t ) ν 4 , φ 2 = ν 2 φ 1 ν 1 , τ 2 ( t ) = − ν 2 τ 3 ( t ) ν 1 , ϕ 1 ( t ) = η 3 , φ 3 ( t ) = 0 , ν 7 ( t ) = ∫ − 2 ( ν 1 2 + ν 4 2 ) ν 3 ( t ) ∫ ν 3 ( t ) d t ν 4 2 d t ,

where η 3 is the integral constant. Substituting Eqs. (7) and (8) into Eq. (2), the interaction solution between lump wave and periodic wave of Eq. (1) is derived as

(9) u = 2 2 ν 1 ∫ ν 3 ( t ) d t + ν 1 x + ν 2 y + 2 ν 4 ν 4 x − ν 1 ∫ ν 3 ( t ) d t ν 4 + ν 2 ν 4 y ν 1 − η 3 φ 1 sin φ 1 x + ν 2 φ 1 y ν 1 ∫ ν 3 ( t ) d t + ν 1 x + ν 2 y 2 + ν 4 x − ν 1 ∫ ν 3 ( t ) d t ν 4 + ν 2 ν 4 y ν 1 2 + η 3 cos φ 1 x + ν 2 φ 1 y ν 1 − ∫ 2 ( ν 1 2 + ν 4 2 ) ν 3 ( t ) ∫ ν 3 ( t ) d t ν 4 2 d t .

The interaction between lump wave and periodic wave is shown in Figure 3. It is easy to see from Figure 3 that with the continuous change of x , a lump wave and a periodic wave propagate forward together, and the amplitude of the lump wave remains unchanged during propagation, which is an elastic collision.

$Figure 3 Solution (9) with ν 1 = 2 {\nu }_{1}=2 , ν 2 = ν 4 = ν 3 ( t ) = η 3 = 1 {\nu }_{2}={\nu }_{4}={\nu }_{3}\left(t)={\eta }_{3}=1 , φ 1 = 3 {\varphi }_{1}=3 , (a) x = − 5 x=-5 , (b) x = 0 x=0 , (c) x = 5 x=5 .$

Figure 3

Solution (9) with ν 1 = 2 , ν 2 = ν 4 = ν 3 ( t ) = η 3 = 1 , φ 1 = 3 , (a) x = − 5 , (b) x = 0 , (c) x = 5 .

4 Breather wave

In order to discuss the breather wave solution of Eq. (1), the following assumptions are usually made

(10) κ = k 3 ( t ) sin ( ς 3 ( t ) + ν 3 x + μ 3 y ) + k 2 ( t ) cos ( ς 2 ( t ) + ν 2 x + μ 2 y ) + k 1 ( t ) e ς 1 ( t ) + ν 1 x + μ 1 y + e − ς 1 ( t ) − ν 1 x − μ 1 y ,

where ν i and μ i ( i = 1 , 2 , 3 ) are unknown constants. k i ( t ) and ς i ( t ) ( i = 1 , 2 , 3 ) are unknown functions. Eq. (10) is different from the breather wave solutions in ref. [11]. Substituting Eq. (10) into Eq. (3), we obtain

(11) ν 3 = − ν 2 , ν 1 = 0 , τ 2 ( t ) = ( μ 3 − μ 2 ) τ 3 ( t ) 2 ν 2 , k 2 ( t ) = k 3 ( t ) = C 1 k 1 ( t ) , ς 1 ( t ) = 1 2 ∫ 2 ν 2 2 μ 1 τ 3 ( t ) − k 1 ′ ( t ) k 1 ( t ) d t , ς 2 ( t ) = 1 2 ν 2 2 ( μ 2 + μ 3 ) ∫ τ 3 ( t ) d t , ς 3 ( t ) = 1 2 ν 2 2 ( μ 2 + μ 3 ) ∫ τ 3 ( t ) d t ,

where C 1 is the integral constant. Substituting Eqs. (10) and (11) into Eq. (2), the breather wave solution of Eq. (1) is presented as

(12) u = − 2 ν 2 C 1 k 1 ( t ) exp ∫ ν 2 2 μ 1 τ 3 ( t ) − k 1 ′ ( t ) 2 k 1 ( t ) d t + μ 1 y × ν 2 x + μ 2 y + sin 1 2 ν 2 2 ( μ 2 + μ 3 ) ∫ τ 3 ( t ) d t + cos − 1 2 ν 2 2 ( μ 2 + μ 3 ) ∫ τ 3 ( t ) d t + ν 2 x − μ 3 y × C 1 k 1 ( t ) exp ∫ ν 2 2 μ 1 τ 3 ( t ) − k 1 ′ ( t ) 2 k 1 ( t ) d t + μ 1 y × cos 1 2 ν 2 2 ( μ 2 + μ 3 ) ∫ τ 3 ( t ) d t + ν 2 x + μ 2 y − sin ν 2 x − μ 3 y − 1 2 ν 2 2 ( μ 2 + μ 3 ) ∫ τ 3 ( t ) d t + k 1 ( t ) exp 2 ∫ ν 2 2 μ 1 τ 3 ( t ) − k 1 ′ ( t ) 2 k 1 ( t ) d t + μ 1 y + 1 .

Figure 4 demonstrates the dynamic properties of breather wave solution when variable coefficient τ 3 ( t ) takes a constant. The dynamic properties of breather wave solution are shown in Figures 5 and 6 when τ 3 ( t ) takes different functions.

$Figure 4 Solution (12) with C 1 = − 2 {C}_{1}=-2 , ν 2 = β 1 = k 1 ( t ) = β 2 = ϑ 6 ( t ) = 1 {\nu }_{2}={\beta }_{1}={k}_{1}\left(t)={\beta }_{2}={{\vartheta }}_{6}\left(t)=1 , β 3 = 0 {\beta }_{3}=0 ; (a) x = − 15 x=-15 , (b) x = 0 x=0 , and (c) x = 15 x=15 .$

Figure 4

Solution (12) with C 1 = − 2 , ν 2 = β 1 = k 1 ( t ) = β 2 = ϑ 6 ( t ) = 1 , β 3 = 0 ; (a) x = − 15 , (b) x = 0 , and (c) x = 15 .

$Figure 5 Solution (12) with C 1 = − 2 {C}_{1}=-2 , ν 2 = μ 1 = k 1 ( t ) = μ 2 = 1 {\nu }_{2}={\mu }_{1}={k}_{1}\left(t)={\mu }_{2}=1 , τ 3 ( t ) = t {\tau }_{3}\left(t)=t , μ 3 = 0 {\mu }_{3}=0 ; (a) x = − 15 x=-15 , (b) x = 0 x=0 , and (c) x = 15 x=15 .$

Figure 5

Solution (12) with C 1 = − 2 , ν 2 = μ 1 = k 1 ( t ) = μ 2 = 1 , τ 3 ( t ) = t , μ 3 = 0 ; (a) x = − 15 , (b) x = 0 , and (c) x = 15 .

$Figure 6 Solution (12) with C 1 = − 2 {C}_{1}=-2 , ν 2 = μ 1 = k 1 ( t ) = μ 2 = 1 {\nu }_{2}={\mu }_{1}={k}_{1}\left(t)={\mu }_{2}=1 , τ 3 ( t ) = sin t {\tau }_{3}\left(t)=\sin t , μ 3 = 0 {\mu }_{3}=0 ; (a) x = − 15 x=-15 , (b) x = 0 x=0 , and (c) x = 15 x=15 .$

Figure 6

Solution (12) with C 1 = − 2 , ν 2 = μ 1 = k 1 ( t ) = μ 2 = 1 , τ 3 ( t ) = sin t , μ 3 = 0 ; (a) x = − 15 , (b) x = 0 , and (c) x = 15 .

5 Conclusion

In this work, we investigate a generalized (2+1)-dimensional variable-coefficient breaking soliton equation. Rich nonautonomous solutions with different wave structures are obtained by using the positive quadratic function method and symbolic computation [13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27]. The breather wave and the interaction among lump wave, solitary wave and periodic wave solutions are studied. The dynamic properties are shown in Figures 1–6. The breather wave and the interaction among lump wave, solitary wave and periodic wave solutions in this work can also be constructed by using the bilinear neural network method [28,29,30]. The assumptions used in this article contain more arbitrary parameters, which can be used to describe more complex physical background and obtain more different forms of solutions. The method is simple, direct and effective. If the bilinear form of a nonlinear partial differential equation can be obtained, the method of this article can be used to solve the equation.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] Liu JG, Ye Q. Stripe solitons and lump solutions for a generalized Kadomtsev-Petviashvili equation with variable coefficients in fluid mechanics. Nonlinear Dyn. 2019;96:23–9. 10.1007/s11071-019-04770-8Search in Google Scholar

[2] Ma WX. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differ Equ. 2018;264:2633–59. 10.1016/j.jde.2017.10.033Search in Google Scholar

[3] Liu JG, Eslami M, Rezazadeh H, Mirzazadeh M. Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev-Petviashvili equation. Nonlinear Dyn. 2019;95(2):1027–33. 10.1007/s11071-018-4612-4Search in Google Scholar

[4] Lan ZZ, Su JJ. Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system. Nonlinear Dyn. 2019;96(4):2535–46. 10.1007/s11071-019-04939-1Search in Google Scholar

[5] Ma WX. Lump solutions to the Kadomtsev-Petviashvili equation. Phys Lett A. 2015;379:1975–8. 10.1016/j.physleta.2015.06.061Search in Google Scholar

[6] Liu JG, Zhu WH, He Y. Variable-coefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients. Z Angew Math Phys. 2021;72:154. 10.1007/s00033-021-01584-wSearch in Google Scholar

[7] Zhang RF, Li MC, Albishari M, Zheng FC, Lan ZZ. Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like equation. Appl Math Comput. 2021;403:126201. 10.1016/j.amc.2021.126201Search in Google Scholar

[8] Ma WX. N-soliton solution of a combined pKP-BKP equation. J Geom Phys. 2021;165:104191. 10.1016/j.geomphys.2021.104191Search in Google Scholar

[9] Liu JG, Wazwaz AM. Breather wave and lump-type solutions of new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid. Math Method Appl Sci. 2021;44(2):2200–8. 10.1002/mma.6931Search in Google Scholar

[10] Zhao C, Gao YT, Lan ZZ, Yang JW. Bäcklund transformation and soliton solutions for a (3+1)-dimensional variable-coefficient breaking soliton equation. Z Naturforsch. 2016;71(9):797–805. 10.1515/zna-2016-0127Search in Google Scholar

[11] Li Q, Shan W, Wang P, Cui H. Breather, lump and N-soliton wave solutions of the (2+1)-dimensional coupled nonlinear partial differential equation with variable coefficients. Commun Nonlinear Sci. 2022;106:106098. 10.1016/j.cnsns.2021.106098Search in Google Scholar

[12] Osman MS. On multi-soliton solutions for the (2+1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide. Comput Math Appl. 2018;75:1–6. 10.1016/j.camwa.2017.08.033Search in Google Scholar

[13] Aasma K, Akmal R, Kottakkaran SN, Osman MS. Splines solutions of boundary value problems that arises in sculpturing electrical process of motors with two rotating mechanism circuit. Phys Scr. 2021;96(10):104001. 10.1088/1402-4896/ac0bd0Search in Google Scholar

[14] Bilge I, Osman MS, Turgut A, Dumitru B. Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations. Math MethodAppl Sci. 2020;43(5):2588–600. 10.1002/mma.6067Search in Google Scholar

[15] Omar AA, Mohammed AS, Hassan A, Dumitru B, Tasawar H, Mohammed A, et al. A novel analytical algorithm for generalized fifth-order time-fractional nonlinear evolution equations with conformable time derivative arising in shallow water waves. Alex Eng J. 2022;61(7):5753–69. 10.1016/j.aej.2021.12.044Search in Google Scholar

[16] Imran S, Mohammed MMJ, Asim Z, Khush BM, Osman MS. Exact traveling wave solutions for two prolific conformable M-Fractional differential equations via three diverse approaches. Results Phys. 2021;28:104557. 10.1016/j.rinp.2021.104557Search in Google Scholar

[17] Sibel T, Karmina KA, Sun TC, Resat Y, Osman MS. Nonlinear pulse propagation for novel optical solitons modeled by Fokas system in monomode optical fibers. Results Phys. 2022;36:105381. 10.1016/j.rinp.2022.105381Search in Google Scholar

[18] Yue C, Elmoasry A, Khater MMA, Osman MS, Attia1 RAM, Lu D, et al. complex wave structures related to the nonlinear long-short wave interaction system: analytical and numerical techniques. AIP Adv. 2020;10(4):045212. 10.1063/5.0002879Search in Google Scholar

[19] Liu JG, Zhu WH, He Y, Lei ZQ. Characteristics of lump solutions to a (3+1)-dimensional variable-coefficient generalized shallow water wave equation in oceanography and atmospheric science. Eur Phys J Plus. 2019;134:385. 10.1140/epjp/i2019-12799-2Search in Google Scholar

[20] Qin CL, Liu JG. Study on double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients. Results Phys. 2022;34(3):105254. 10.1016/j.rinp.2022.105254Search in Google Scholar

[21] Liu JG, Wazwaz AM, Zhu WH. Solitary and lump waves interaction in variable-coefficient nonlinear evolution equation by a modified ansätz with variable coefficients. J Appl Anal Comput. 2022;12(2):517–32. 10.11948/20210178Search in Google Scholar

[22] Sachin K, Shubham KD, Dumitru B, Osman MS, Wazwaz AM. Lie symmetries, closed-form solutions, and various dynamical profiles of solitons for the variable coefficient (2+1)-dimensional KP equations. Symmetry. 2022;14(3):597. 10.3390/sym14030597Search in Google Scholar

[23] Liu JG, Zhu WH, Zhou L. Interaction solutions for Kadomtsev-Petviashvili equation with variable coefficients. Commun Theor Phys. 2019;71:793–7. 10.1088/0253-6102/71/7/793Search in Google Scholar

[24] Karmina KA, Resat Y, Osman MS. Dynamic behavior of the (3+1)-dimensional KdV-Calogero-Bogoyavlenskii-Schiff equation. Opt Quant Electron. 2022;54(3):160. 10.1007/s11082-022-03528-8Search in Google Scholar

[25] Liu JG, Osman MS. Nonlinear dynamics for different nonautonomous wave structures solutions of a 3D variable-coefficient generalized shallow water wave equation. Chinese J Phys. 2022;72:1618–24. 10.1016/j.cjph.2021.10.026Search in Google Scholar

[26] Asim Z, Muhammad R, Muhammad QZ, Kottakkaran SN, Osman MS, Roshan NM, et al. Dynamics of different nonlinearities to the perturbed nonlinear Schrödinger equation via solitary wave solutions with numerical simulation. Fractal Fract. 2021;5(4):213. 10.3390/fractalfract5040213Search in Google Scholar

[27] Liu JG, Zhu WH, Osman MS, Ma WH. An explicit plethora of different classes of interactive lump solutions for an extension form of 3D–Jimbo-Miwa model. Eur Phys J Plus. 2020;135(5):412. 10.1140/epjp/s13360-020-00405-9Search in Google Scholar

[28] Zhang LF, Li MC. Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn. 2022;108:521–31. 10.1007/s11071-022-07207-xSearch in Google Scholar

[29] Zhang LF, Li MC, Yin HM. Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo-Miwa equation. Nonlinear Dyn. 2021;103:1071–9. 10.1007/s11071-020-06112-5Search in Google Scholar

[30] Zhang LF, Sudao B. Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 2019;95:3041–8. 10.1007/s11071-018-04739-zSearch in Google Scholar

Received: 2022-03-04

Revised: 2022-04-02

Accepted: 2022-05-27

Published Online: 2022-06-18

This work is licensed under the Creative Commons Attribution 4.0 International License.

Nonlinear dynamics for different nonautonomous wave structure solutions

Abstract

1 Introduction

2 Interaction between lump wave and solitary wave

3 Interaction between lump wave and periodic wave

4 Breather wave

5 Conclusion

References

Journal and Issue

Articles in the same Issue