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A new (3 + 1)-dimensional Schrödinger equation: derivation, soliton solutions and conservation laws

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Abstract

In the present paper, a new (3 + 1)-dimensional Schrödinger equation in Quantum Mechanics is derived. Based on the extended (3 + 1)-dimensional zero curvature equation, this equation is derived for the first time via the compatibility condition. Meanwhile, some soliton solutions are presented. Finally, conservation laws also obtained.

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References

  1. Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)

    Book  Google Scholar 

  2. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  3. Rogers, C., Schief, W.K.: B\(\ddot{a}\)cklund and Darboux Transformations, Geometry and Morden Applications in Soliton Theory. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  4. Gu, C.H., Hu, H.S., et al.: Darboux Transformations in Integrable Systems Theory and Their Applications to Geometry. Springer, Berlin (2005)

    MATH  Google Scholar 

  5. Hu, X.B., Zhu, Z.N.: A B\(\ddot{a}\)cklund transformation and nonlinear superposition formula for the Belov–Chaltikian lattice. J. Phys. A. 31, 4716–4755 (1998)

    Google Scholar 

  6. Hu, W.P., et al.: Symmetry breaking of infinite-dimensional dynamic system. Appl. Math. Lett. 103, 106207 (2020)

    Article  MathSciNet  Google Scholar 

  7. Hu, W.P., Zhang, C.Z., Deng, Z.C.: Vibration and elastic wave propagation in spatial flexible damping panel attached to four special springs. Commun. Nonlinear Sci. Numer. Simul. 84, 105199 (2020)

    Article  MathSciNet  Google Scholar 

  8. Hu, W.P., Ye, J., Deng, Z.C.: Internal resonance of a flexible beam in a spatial tethered system. J. Sound Vib. 475, 115286 (2020)

    Article  Google Scholar 

  9. Hu, W.P., et al.: Coupling dynamic behaviors of flexible stretching hub-beam system. Mech. Syst. Signal Proc. 151, 107389 (2021)

    Article  Google Scholar 

  10. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)

    Article  MathSciNet  Google Scholar 

  11. Olver, P.J.: Application of Lie Group to Differential Equation. Springer, New York (1986)

    Book  Google Scholar 

  12. Bluman, G.W., Cheviakov, A., Anco, S.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2010)

    Book  Google Scholar 

  13. Wang, G.W., et al.: A (2 + 1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions. Nucl. Phys. B 953, 114956 (2020)

    Article  MathSciNet  Google Scholar 

  14. Wang, G.W., et al.: (2 + 1)-Dimensional Boiti–Leon–Pempinelli equation—domain walls, invariance properties and conservation laws. Phys. Lett. A 384, 126255 (2020)

    Article  MathSciNet  Google Scholar 

  15. Wang, G.W., et al.: Symmetry analysis for a seventh-order generalized KdV equation and its fractional version in fluid mechanics. Fractals 28, 2050044 (2020)

    Article  Google Scholar 

  16. Wang, G.W.: Symmetry analysis and rogue wave solutions for the (2 + 1)-dimensional nonlinear Schrödinger equation with variable coefficients. Appl. Math. Lett. 56, 56–64 (2016)

    Article  MathSciNet  Google Scholar 

  17. Wang, G.W., Kara, A.H.: A (2 + 1)-dimensional KdV equation and mKdV equation: symmetries, group invariant solutions and conservation laws. Phys. Lett. A 383, 728–731 (2019)

    Article  MathSciNet  Google Scholar 

  18. Wang, G.W.: Symmetry analysis, analytical solutions and conservation laws of a generalized KdV–Burgers–Kuramoto equation and its fractional version. Fractals (2021). https://doi.org/10.1142/S0218348X21501012

    Article  Google Scholar 

  19. Muatjetjeja, B., Mbusi, S.O., Adem, A.R.: Noether symmetries of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry. Symmetry 12, 566 (2020)

    Article  Google Scholar 

  20. Adem, A.R.: The generalized (1 + 1)-dimensional and (2 + 1)-dimensional Ito equations: multiple exp-function algorithm and multiple wave solutions. Comput. Math. Appl. 71, 1248–1258 (2016)

    Article  MathSciNet  Google Scholar 

  21. Muatjetjeja, B., Adem, A.R., Mbusi, S.O.: Traveling wave solutions and conservation laws of a generalized Kudryashov–Sinelshchikov equation. J. Appl. Anal. 25, 211–217 (2019)

    Article  MathSciNet  Google Scholar 

  22. Wang, G.W.: A novel (3 + 1)-dimensional sine-Gorden and sinh-Gorden equation: derivation, symmetries and conservation laws. Appl. Math. Lett. 113, 106768 (2021)

    Article  MathSciNet  Google Scholar 

  23. Zhou, Z.: Finite dimensional Hamiltonians and almost-periodic solutions for (2 + 1) dimensional three-wave equations. J. Phys. Soc. Jpn. 71, 1857–1863 (2002)

    Article  MathSciNet  Google Scholar 

  24. Latha, M.M., Vasanthi, C.C.: An integrable model of (2 + 1)-dimensional Heisenberg ferromagnetic spin chain and soliton excitations. Phys. Scr. 89, 065204 (2014)

    Article  Google Scholar 

  25. Biswas, A., et al.: Optical solitons and complexitons of the Schrodinger–Hirota equation. Opt. Laser Technol. 44, 2265–2269 (2012)

    Article  Google Scholar 

  26. Tang, G.S., Wang, S.H., Wang, G.W.: Solitons and complexitons solutions of an integrable model of (2 + 1)-dimensional Heisenberg ferromagnetic spin chain. Nonlinear Dyn. 88, 2319–2327 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by Natural Science Foundation of Hebei Province of China (No. A2018207030), Youth Key Program of Hebei University of Economics and Business (2018QZ07), Key Program of Hebei University of Economics and Business (2020ZD11), Youth Team Support Program of Hebei University of Economics and Business.

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  1. School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, People’s Republic of China

    Gangwei Wang

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  1. Gangwei Wang
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Correspondence to Gangwei Wang.

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Wang, G. A new (3 + 1)-dimensional Schrödinger equation: derivation, soliton solutions and conservation laws. Nonlinear Dyn 104, 1595–1602 (2021). https://doi.org/10.1007/s11071-021-06359-6

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