Abstract
The linear superposition principle is applied to hyperbolic and trigonometric function solutions to generalized bilinear equations. We determine sufficient and necessary conditions for the existence of linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations. By using weights, three examples are given to show applicability of our theory.
Funding statement: Funding: The work was supported in part by NNSFC under the grants 11371326, 11271008, and 61072147, Natural Science Foundation of Shanghai (Grant No. 11ZR1414100), Zhejiang Innovation Project of China (Grant No. T200905), and the First-class Discipline of Universities in Shanghai and the Shanghai University Leading Academic Discipline Project (No. A.13-0101-12-004).
Acknowledgements:
The authors are also grateful to R. Dougherty, X. Gu, X. Lyu, S. Manukure, M. Mcanally, Y.J. Zhang, Y. Zhou, for their valuable discussions in the differential equation seminar at University of South Florida. The authors also thank “Republic of Turkey-Presidency of Higher Education” for providing a study abroad scholarship to Ö. Ünsal.
References
[1] Ablowitz M.J. and Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991.10.1017/CBO9780511623998Suche in Google Scholar
[2] Zakharov V.E., What is integrability? Springer-Verlag, Berlin, 1991.10.1007/978-3-642-88703-1Suche in Google Scholar
[3] Ma W.X., Integrability, in: Encyclopedia of nonlinear science (SCOTT, A. ed.), pp. 450–453, Taylor & Francis, New York, 2005.Suche in Google Scholar
[4] Lü X., Ma W.X., Yu J. and Khalique C.M., Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 31 (2016), 40.10.1016/j.cnsns.2015.07.007Suche in Google Scholar
[5] Lü X., Lin F., Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order, Commun. Nonlinear Sci. Numer. Simul. 32 (2016), 241.10.1016/j.cnsns.2015.08.008Suche in Google Scholar
[6] Lü X., Ma W.X. and Khalique C.M., A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg-de Vries-like model, Appl. Math. Lett. 50 (2015), 37.10.1016/j.aml.2015.06.003Suche in Google Scholar
[7] Lü X., Madelung fluid description on a generalized mixed nonlinear Schrödinger equation, Nonlinear Dyn. 81 (2015), 239.10.1007/s11071-015-1985-5Suche in Google Scholar
[8] Lü X., Lin F. and Qi F., Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions, Appl. Math. Modell. 39 (2015), 3221.10.1016/j.apm.2014.10.046Suche in Google Scholar
[9] Lü X., Ma W.X., Yu J., Lin F. and Khalique C.M., Envelope bright- and dark-soliton solutions for the Gerdjikov-Ivanov model, Nonlinear Dyn. 82 (2015), 1211.10.1007/s11071-015-2227-6Suche in Google Scholar
[10] Hirota R., The direct method in soliton theory, Cambridge University Press, New York, 2004.10.1017/CBO9780511543043Suche in Google Scholar
[11] Hietarinta J., Hirota’s bilinear method and soliton solutions, Phys. AUC 15 (2005), 31.Suche in Google Scholar
[12] Wazwaz A.M., Solitary wave solutions for a K(m,n,p,q+r) equation with generalized evolution, Int. J. Nonlinear Sci. 12 (2011), 471.Suche in Google Scholar
[13] Zheng H.C., Ma W.X. and Gu X., Hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions, Appl. Math. Comput. 220 (2013), 226.10.1016/j.amc.2013.06.019Suche in Google Scholar
[14] Ma W.X. and Fan E.G., Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61 (2011), 950.10.1016/j.camwa.2010.12.043Suche in Google Scholar
[15] Ma W.X., Zhang Y., Tang Y.N. and Tu J.Y., Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012), 7174.10.1016/j.amc.2011.12.085Suche in Google Scholar
[16] Ma W.X., Generalized bilinear differential equations, Stud. Nonlinear Sci. 2 (2011), 140.Suche in Google Scholar
[17] Ma W.X., Bilinear equations, Bell polynomials and linear superposition principle, J. Phys. Conf. Ser. 411 (2013), 012021.10.1088/1742-6596/411/1/012021Suche in Google Scholar
[18] Ma W.X., Bilinear equations and resonant solutions characterized by Bell polynomials, Rep. Math. Phys. 72 (2013), 41.10.1016/S0034-4877(14)60003-3Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation
- Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn–Hilliard Equation
- On the Mittag–Leffler Stability of Impulsive Fractional Solow-Type Models
- Non-similarity Solutions for Viscous Dissipation and Soret Effects in Micropolar Fluid over a Truncated Cone with Convective Boundary Condition: Spectral Quasilinearization Approach
- Numerical Simulation of the Supersonic Disk-Gap-Band Parachute by Using Implicit Coupling Method
- Stochastic-Based RANS-LES Simulations of Swirling Turbulent Jet Flows
- Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections
- Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients
- Multiple-Wave Solutions to Generalized Bilinear Equations in Terms of Hyperbolic and Trigonometric Solutions
- Experimental and Simulation Analysis of the Successful Production of Heavy-Gauge Steel Plate by the Clad Rolling Process
- Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations
- A Note on Hidden Transient Chaos in the Lorenz System
- Finite Time Blow-up in a Delayed Diffusive Population Model with Competitive Interference
Artikel in diesem Heft
- Frontmatter
- The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation
- Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn–Hilliard Equation
- On the Mittag–Leffler Stability of Impulsive Fractional Solow-Type Models
- Non-similarity Solutions for Viscous Dissipation and Soret Effects in Micropolar Fluid over a Truncated Cone with Convective Boundary Condition: Spectral Quasilinearization Approach
- Numerical Simulation of the Supersonic Disk-Gap-Band Parachute by Using Implicit Coupling Method
- Stochastic-Based RANS-LES Simulations of Swirling Turbulent Jet Flows
- Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections
- Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients
- Multiple-Wave Solutions to Generalized Bilinear Equations in Terms of Hyperbolic and Trigonometric Solutions
- Experimental and Simulation Analysis of the Successful Production of Heavy-Gauge Steel Plate by the Clad Rolling Process
- Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations
- A Note on Hidden Transient Chaos in the Lorenz System
- Finite Time Blow-up in a Delayed Diffusive Population Model with Competitive Interference