Geodesic
Exact solutions of the modified Korteweg–de Vries equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 35-48 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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F. Demontis. Exact solutions of the modified Korteweg–de Vries equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 35-48. http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a3/

[1] S. Matsutani, H. Tsuru, J. Phys. Soc. Japan, 60:11 (1991), 3640–3644 | DOI | MR

[2] H. Ono, J. Phys. Soc. Japan, 61:12 (1992), 4336–4343 | DOI

[3] E. A. Ralph, L. Pratt, J. Nonlinear Sci., 4:1 (1994), 355–374 | DOI | Zbl

[4] T. S. Komatsu, S.-I. Sasa, Phys. Rev. E, 52:5 (1995), 5574–5582, arXiv: patt-sol/9506002 | DOI

[5] T. Nagatani, Physica A, 265:1–2 (1999), 297–310 | DOI

[6] Z.-P. Li, Y.-C. Liu, Eur. Phys. J. B, 53:3 (2006), 367–374 | DOI | Zbl

[7] H. X. Ge, S. Q. Dai, Y. Xue, L. Y. Dong, Phys. Rev. E, 71:6 (2005), 066119, 7 pp. | DOI | MR

[8] W. K. Schief, Nonlinearity, 8:1 (1995), 1–9 | DOI | MR | Zbl

[9] K. E. Lonngren, Opt. Quant. Electron., 30:7–10 (1998), 615–630 | DOI

[10] A. H. Khater, O. H. El-Kalaawy, D. K. Callebaut, Phys. Scr., 58:6 (1998), 545–548 | DOI

[11] M. Agop, V. Cojocaru, Mater. Trans., 39:6 (1998), 668–671 | DOI

[12] V. Ziegler, J. Dinkel, C. Setzer, K. E. Lonngren, Chaos, Solitons and Fractals, 12:9 (2001), 1719–1728 | DOI | Zbl

[13] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser., 149, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[14] M. Ablovits, Kh. Sigur, Solitony i metod obratnoi zadachi, Mir, M., 1987 | MR | MR | Zbl

[15] M. J. Ablowitz, B. Prinari, A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Math. Soc. Lecture Note Ser., 302, Cambridge Univ. Press, Cambridge, 2004 | MR | Zbl

[16] Dzh. L. Lem, Vvedenie v teoriyu solitonov, Mir, M., 1983 | MR | MR | Zbl

[17] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR | MR | Zbl

[18] V. E. Zakharov, A. B. Shabat, ZhETF, 61:1 (1971), 118–134 | MR

[19] T. Aktosun, C. van der Mee, Inverse Problems, 22:6 (2006), 2165–2174, arXiv: math-ph/0606008 | DOI | MR | Zbl

[20] T. Aktosun, F. Demontis, C. van der Mee, Inverse Problems, 23:5 (2007), 2171–2195, arXiv: nlin/0702010 | DOI | MR | Zbl

[21] T. Aktosun, F. Demontis, C. van der Mee, J. Math. Phys., 51:12 (2010), 123521, 27 pp., arXiv: 1003.2453 | DOI | MR | Zbl

[22] F. Demontis, C. van der Mee, Inverse Problems, 24:2 (2008), 025020, 16 pp. | DOI | MR | Zbl

[23] H. Ono, J. Phys. Soc. Japan, 41:5 (1976), 1817–1818 | DOI | MR | Zbl

[24] M. Wadati, J. Phys. Soc. Japan, 34:5 (1973), 1289–1296 | DOI | MR | Zbl

[25] F. Demontis, Direct and inverse scattering of the matrix Zakharov–Shabat system, Ph. D. thesis, Univ. of Cagliari, Italy, 2007

[26] C. van der Mee, “Direct and inverse scattering for skewselfadjoint Hamiltonian systems”, Current Trends in Operator Theory and its Applications, Oper. Theory Adv. Appl., 149, eds. J. A. Ball, J. W. Helton, M. Klaus, L. Rodman, Birkhäuser, Basel, 2004, 407–439 | MR | Zbl

[27] T. N. Busse, Generalized Inverse Scattering Transform for the Nonlinear Schrödinger Equation, Ph. D. thesis, Univ. of Texas, Arlington, TX, 2008 | MR

[28] H. Bart, I. Gohberg, M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Oper. Theory Adv. Appl., 1, Birkhäuser, Basel, 1979 | DOI | MR | Zbl

[29] H. Dym, Linear Algebra in Action, Graduate Stud. Math., 78, AMS, Providence, RI, 2007 | MR | Zbl