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The Initial-Boundary-Value Problems for the Hirota Equation on the Half-Line

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Abstract

An initial boundary-value problem for the Hirota equation on the half-line, 0 < x < ∞, t > 0, is analysed by expressing the solution q(x, t) in terms of the solution of a matrix Riemann-Hilbert (RH) problem in the complex k-plane. This RH problem has explicit (x, t) dependence and it involves certain functions of k referred to as the spectral functions. Some of these functions are defined in terms of the initial condition q(x, 0) = q0(x), while the remaining spectral functions are defined in terms of the boundary values q(0, t) = g0(t), qx(0, t) = g1(t) and qxx(0, t) = g2(t). The spectral functions satisfy an algebraic global relation which characterizes, say, g2(t) in terms of {q0(x), g0(t), g1(t)}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

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References

  1. Hirota, R., Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14, 1973, 805–809.

    Article  MathSciNet  Google Scholar 

  2. Lakshmanan, M. and Ganesan, S., Equivalent forms of a generalized Hirota’s equation with linear inhomogeneities, J. Phys. Soc. Jpn., 52, 1983, 4031–4034.

    Article  MathSciNet  Google Scholar 

  3. Lamb, G. L., Elements of Soliton Theory, Wiley, New York, 1980.

    MATH  Google Scholar 

  4. Tao, Y. and He, J., Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation, Phys. Rev. E, 85, 2012, 026601.

    Article  Google Scholar 

  5. Fokas, A. S., A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. Lond. A, 453, 1997, 1411–1443.

    Article  MathSciNet  Google Scholar 

  6. Fokas, A. S., Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230, 2002, 1–39.

    Article  MathSciNet  Google Scholar 

  7. Fokas, A. S., A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2008.

    Book  Google Scholar 

  8. Lenells, J., Initial-boundary value problems for integrable evolution equations with 3 × 3 Lax pairs, Physica D, 241, 2012, 857–875.

    Article  MathSciNet  Google Scholar 

  9. Lenells, J., The KdV equation on the half-line: The Dirichlet to Neumann map, J. Phys. A, 46, 2013, 345203.

    Article  MathSciNet  Google Scholar 

  10. Lenells, J., The Degasperis-Procesi equation on the half-line, Nonlinear Analysis, 76, 2013, 122–139.

    Article  MathSciNet  Google Scholar 

  11. Boutet de Monvel, A., Fokas, A. S. and Shepelsky, D., The mKDV equation on the half-line, J. Inst. Math. Jussieu., 3, 2004, 139–164.

    Article  MathSciNet  Google Scholar 

  12. Boutet de Monvel, A. and Shepelsky, D., The Camassa-Holm equation on the half-line: A Riemann-Hilbert Approach, J. Geom. Anal., 18, 2008, 285–323.

    Article  MathSciNet  Google Scholar 

  13. Xu, J. and Fan, E., The unified transform method for the Sasa-Satsuma equation on the half-line, Proc. R. Soc. A, 469, 2013, 20130068.

    Article  MathSciNet  Google Scholar 

  14. Xu, J. and Fan, E., The three wave equation on the half-line, Physics Letters A, 378, 2014, 26–33.

    Article  MathSciNet  Google Scholar 

  15. Bona, J. L. and Fokas, A. S., Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equations, Nonlinearity, 21, 2008, T195–T203.

    Article  MathSciNet  Google Scholar 

  16. Boutet De Monvel, A., Fokas, A. S. and Shepelsky, D., Integrable nonlinear evolution equations on a finite interval, Comm.. Math. Phys., 263, 2006, 133–172.

    Article  MathSciNet  Google Scholar 

  17. Dujardin, G. M., Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals, Proc. R. Soc. Lond. A, 465, 2009, 3341–3360.

    Article  MathSciNet  Google Scholar 

  18. Fokas, A. S., Integrable nonlinear evolution equations on the half-line, Comm.. Math. Phys., 230, 2002, 1–39.

    Article  MathSciNet  Google Scholar 

  19. Fokas, A. S., Linearizable initial boundary value problems for the sine-Gordon equation on the half-line, Nonlinearity, 17, 2004, 1521–1534.

    Article  MathSciNet  Google Scholar 

  20. Fokas, A. S. and Its, A. R., The Nonlinear Schrödinger equation on the interval, J. Phys. A, 37, 2004, 6091–6114.

    Article  MathSciNet  Google Scholar 

  21. Fokas, A. S., Its, A. R. and Sung, L. Y., The nonlinear Schrodinger equation on the half-line, Nonlinearity, 18, 2005, 1771–1822.

    Article  MathSciNet  Google Scholar 

  22. Lenells, J., The derivative nonlinear Schrodinger equation on the half-line, Physica D, 237, 2008, 3008–3019.

    Article  MathSciNet  Google Scholar 

  23. Fokas, A. S. and Its, A. R., The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal., 27, 1996, 738–764.

    Article  MathSciNet  Google Scholar 

  24. Zakharov, V. E. and Shabat, A. B., A scheme for integrating the nonlinear equations of numerical physics by the method of the inverse scattering problem I, Funct. Anal. Appl., 8, 1974, 226–235.

    Article  Google Scholar 

  25. Zakharov, V. E. and Shabat, A. B., A scheme for integrating the nonlinear equations of numerical physics by the method of the inverse scattering problem II, Funct. Anal. Appl., 13, 1979, 166–174.

    Article  Google Scholar 

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Acknowledgment

The author sincerely thank Prof. Engui Fan and Dr. Jian Xu for many useful discussions.

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Authors and Affiliations

  1. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Lin Huang

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Lin Huang

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  1. Lin Huang
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Correspondence to Lin Huang.

Additional information

This work was supported by the China Postdoctoral Science Foundation (No. 2015M580285).

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Huang, L. The Initial-Boundary-Value Problems for the Hirota Equation on the Half-Line. Chin. Ann. Math. Ser. B 41, 117–132 (2020). https://doi.org/10.1007/s11401-019-0189-6

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  • DOI: https://doi.org/10.1007/s11401-019-0189-6

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