Abstract
Based on the positive quadratic function method, the rich nonautonomous solutions of a generalized (2+1)dimensional variablecoefficient 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.
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 highdimensional nonlinear development equations using this method [6,7, 8,9]. The research of these solutions has important significance and prospects for many highdimensional nonlinear problems in mathematics, physics and other fields.
Variable coefficient integrability systems can more clearly describe realworld 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 largescale forward changes in ocean temperature–salinity relationship and other dynamic conditions [10]. In this article, under investigation is the generalized (2+1)dimensional variablecoefficient breaking soliton equation [11]
where
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
Eq. (2) becomes
In order to investigate the interaction between lump wave and solitary wave of Eq. (1), the following assumptions are usually made
where
where
When
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
Substituting Eq. (7) into Eq. (3), we obtain
where
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
4 Breather wave
In order to discuss the breather wave solution of Eq. (1), the following assumptions are usually made
where
where
Figure 4 demonstrates the dynamic properties of breather wave solution when variable coefficient
5 Conclusion
In this work, we investigate a generalized (2+1)dimensional variablecoefficient 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 KadomtsevPetviashvili equation with variable coefficients in fluid mechanics. Nonlinear Dyn. 2019;96:23–9. 10.1007/s11071019047708Search 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 nonisospectral and generalized variablecoefficient KadomtsevPetviashvili equation. Nonlinear Dyn. 2019;95(2):1027–33. 10.1007/s1107101846124Search in Google Scholar
[4] Lan ZZ, Su JJ. Solitary and rogue waves with controllable backgrounds for the nonautonomous generalized AB system. Nonlinear Dyn. 2019;96(4):2535–46. 10.1007/s11071019049391Search in Google Scholar
[5] Ma WX. Lump solutions to the KadomtsevPetviashvili 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. Variablecoefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients. Z Angew Math Phys. 2021;72:154. 10.1007/s0003302101584wSearch 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 CaudreyDoddGibbonKoteraSawadalike equation. Appl Math Comput. 2021;403:126201. 10.1016/j.amc.2021.126201Search in Google Scholar
[8] Ma WX. Nsoliton solution of a combined pKPBKP equation. J Geom Phys. 2021;165:104191. 10.1016/j.geomphys.2021.104191Search in Google Scholar
[9] Liu JG, Wazwaz AM. Breather wave and lumptype solutions of new (3+1)dimensional BoitiLeonMannaPempinelli 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 variablecoefficient breaking soliton equation. Z Naturforsch. 2016;71(9):797–805. 10.1515/zna20160127Search in Google Scholar
[11] Li Q, Shan W, Wang P, Cui H. Breather, lump and Nsoliton 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 multisoliton solutions for the (2+1)dimensional breaking soliton equation with variable coefficients in a gradedindex 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. 10.1088/14024896/ac0bd0Search in Google Scholar
[14] Bilge I, Osman MS, Turgut A, Dumitru B. Analytical and numerical solutions of mathematical biology models: The NewellWhiteheadSegel and AllenCahn 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 fifthorder timefractional 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 MFractional 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 longshort 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 variablecoefficient generalized shallow water wave equation in oceanography and atmospheric science. Eur Phys J Plus. 2019;134:385. 10.1140/epjp/i2019127992Search in Google Scholar
[20] Qin CL, Liu JG. Study on doubleperiodic soliton and nontraveling 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 variablecoefficient 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, closedform 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 KadomtsevPetviashvili equation with variable coefficients. Commun Theor Phys. 2019;71:793–7. 10.1088/02536102/71/7/793Search in Google Scholar
[24] Karmina KA, Resat Y, Osman MS. Dynamic behavior of the (3+1)dimensional KdVCalogeroBogoyavlenskiiSchiff equation. Opt Quant Electron. 2022;54(3):160. 10.1007/s11082022035288Search in Google Scholar
[25] Liu JG, Osman MS. Nonlinear dynamics for different nonautonomous wave structures solutions of a 3D variablecoefficient 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–JimboMiwa model. Eur Phys J Plus. 2020;135(5):412. 10.1140/epjp/s13360020004059Search 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/s1107102207207xSearch in Google Scholar
[29] Zhang LF, Li MC, Yin HM. Rogue wave solutions and the bright and dark solitons of the (3+1)dimensional JimboMiwa equation. Nonlinear Dyn. 2021;103:1071–9. 10.1007/s11071020061125Search 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 pgBKP equation. Nonlinear Dyn. 2019;95:3041–8. 10.1007/s1107101804739zSearch in Google Scholar
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