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TMF, 2011, Volume 168, Number 1, Pages 35–48 (Mi tmf6662)  

This article is cited in 12 scientific papers (total in 12 papers)

Exact solutions of the modified Korteweg–de Vries equation

F. Demontis

Dipartimento Matematica, Università di Cagliari, Cagliari, Italy

Abstract: We use the inverse scattering method to obtain a formula for certain exact solutions of the modified Korteweg–de Vries (mKdV) equation. Using matrix exponentials, we write the kernel of the relevant Marchenko integral equation as $\Omega(x+y;t)=Ce^{-(x+y)A}e^{8A^3 t}B$, where the real matrix triplet $(A,B,C)$ consists of a constant $p{\times}p$ matrix $A$ with eigenvalues having positive real parts, a constant $p\times1$ matrix $B$, and a constant $1\times p$ matrix $C$ for a positive integer $p$. Using separation of variables, we explicitly solve the Marchenko integral equation, yielding exact solutions of the mKdV equation. These solutions are constructed in terms of the unique solution $P$ of the Sylvester equation $AP+PA=BC$ or in terms of the unique solutions $Q$ and $N$ of the Lyapunov equations $A^\dag Q+QA=C^\dag C$ and $AN+NA^\dag=BB^\dag$, where $B^\dag$ denotes the conjugate transposed matrix. We consider two interesting examples.

Keywords: inverse scattering method, Lyapunov equation, explicit solution of the modified Korteweg–de Vries equation


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English version:
Theoretical and Mathematical Physics, 2011, 168:1, 886–897

Bibliographic databases:

Citation: F. Demontis, “Exact solutions of the modified Korteweg–de Vries equation”, TMF, 168:1 (2011), 35–48; Theoret. and Math. Phys., 168:1 (2011), 886–897

Citation in format AMSBIB
\by F.~Demontis
\paper Exact solutions of the~modified Korteweg--de Vries equation
\jour TMF
\yr 2011
\vol 168
\issue 1
\pages 35--48
\jour Theoret. and Math. Phys.
\yr 2011
\vol 168
\issue 1
\pages 886--897

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    This publication is cited in the following articles:
    1. Zhang D.-J., Zhao S.-L., Sun Y.-Y., Zhou J., “Solutions To the Modified Korteweg-de Vries Equation”, Rev. Math. Phys., 26:7 (2014), 1430006  crossref  mathscinet  zmath  isi  scopus
    2. Xu D.-d., Zhang D.-j., Zhao S.-l., “The Sylvester Equation and Integrable Equations: I. The Korteweg-de Vries System and sine-Gordon Equation”, J. Nonlinear Math. Phys., 21:3 (2014), 382–406  crossref  mathscinet  isi  scopus
    3. Demontis F., Ortenzi G., van der Mee C., “Exact Solutions of the Hirota Equation and Vortex Filaments Motion”, Physica D, 313 (2015), 61–80  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Guner O., Aksoy E., Bekir A., Cevikel A.C., “Different methods for $(3+1)$-dimensional space-time fractional modified KdV-Zakharov–Kuznetsov equation”, Comput. Math. Appl., 71:6 (2016), 1259–1269  crossref  mathscinet  isi  scopus
    5. Sahoo S., Garai G., Ray S.S., “Lie symmetry analysis for similarity reduction and exact solutions of modified KdV-Zakharov–Kuznetsov equation”, Nonlinear Dyn., 87:3 (2017), 1995–2000  crossref  mathscinet  zmath  isi  scopus
    6. Lu D., Seadawy A.R., Arshad M., Wang J., “New Solitary Wave Solutions of (3”, Results Phys., 7 (2017), 899–909  crossref  isi  scopus
    7. Guner O., “New Exact Solutions to the Space-Time Fractional Nonlinear Wave Equation Obtained By the Ansatz and Functional Variable Methods”, Opt. Quantum Electron., 50:1 (2018), 38  crossref  mathscinet  isi  scopus
    8. Demontis F., Ortenzi G., van der Mee C., “Nonlocal Integrable PDEs From Hierarchies of Symmetry Laws: the Example of Pohlmeyer-Lund-Regge Equation and Its Reflectionless Potential Solutions”, J. Geom. Phys., 127 (2018), 84–100  crossref  mathscinet  zmath  isi  scopus
    9. Demontis F., Ortenzi G., Sommacal M., “Heisenberg Ferromagnetism as An Evolution of a Spherical Indicatrix: Localized Solutions and Elliptic Dispersionless Reduction”, Electron. J. Differ. Equ., 2018, 106  zmath  isi
    10. Demontis F., Lombardo S., Sommacal M., van der Mee C., Vargiu F., “Effective Generation of Closed-Form Soliton Solutions of the Continuous Classical Heisenberg Ferromagnet Equation”, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 35–65  crossref  mathscinet  isi  scopus
    11. Demontis F., Ortenzi G., Sommacal M., van der Mee C., “The Continuous Classical Heisenberg Ferromagnet Equation With in-Plane Asymptotic Conditions. i. Direct and Inverse Scattering Theory”, Ric. Mat., 68:1 (2019), 145–161  crossref  isi
    12. Demontis F., Ortenzi G., Sommacal M., van der Mee C., “The Continuous Classical Heisenberg Ferromagnet Equation With in-Plane Asymptotic Conditions. II. Ist and Closed-Form Soliton Solutions”, Ric. Mat., 68:1 (2019), 163–178  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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