Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
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- by Wen-Xiu Ma and Yuncheng You PDF
- Trans. Amer. Math. Soc. 357 (2005), 1753-1778 Request permission
Abstract:
A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.References
- Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. MR 450815, DOI 10.1002/sapm1974534249
- M. J. Ablowitz and J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys. 19 (1978), no. 10, 2180–2186. MR 507515, DOI 10.1063/1.523550
- Mark J. Ablowitz and Harvey Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 642018, DOI 10.1137/1.9781611970883
- M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), no. 1, 1–30. MR 501106, DOI 10.1007/BF01609465
- V. A. Arkad′ev, A. K. Pogrebkov, and M. K. Polivanov, Singular solutions of the KdV equation and the method of the inverse problem, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 133 (1984), 17–37 (Russian, with English summary). Differential geometry, Lie groups and mechanics, VI. MR 742146
- D. K. Arrowsmith and C. M. Place, Dynamical systems, Chapman and Hall Mathematics Series, Chapman & Hall, London, 1992. Differential equations, maps and chaotic behaviour. MR 1195127, DOI 10.1007/978-94-011-2388-4
- D. J. Benney and D. J. Roskes, Wave instabilities, Studies in Appl. Math. 48 (1969), 377–385.
- A. C. Bryan and A. E. G. Stuart, Representations of the multisoliton solutions of the Korteweg-de Vries equation, Nonlinear Anal. 22 (1994), no. 5, 561–566. MR 1266543, DOI 10.1016/0362-546X(94)90082-5
- Robin K. Bullough and Philip J. Caudrey (eds.), Solitons, Topics in Current Physics, vol. 17, Springer-Verlag, Berlin-New York, 1980. MR 625877
- Nigel J. Burroughs, A loop algebra co-adjoint orbit construction of the generalized KdV hierarchies, Nonlinearity 6 (1993), no. 4, 583–616. MR 1231775, DOI 10.1088/0951-7715/6/4/005
- A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974), 101–110. MR 349126, DOI 10.1098/rspa.1974.0076
- P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322, DOI 10.1017/CBO9781139172059
- N. C. Freeman and J. J. C. Nimmo, Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Phys. Lett. A 95 (1983), no. 1, 1–3. MR 700477, DOI 10.1016/0375-9601(83)90764-8
- R. Hirota, Exact solution of the Korteweg de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192–1194.
- Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0486784
- M. Jaworski, Breather-like solution of the Korteweg-de Vries equation, Phys. Lett. A 104 (1984), no. 5, 245–247. MR 758224, DOI 10.1016/0375-9601(84)90060-4
- B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl. 15 (1970), 539–541.
- M. Kovalyov, Basic motions of the Korteweg-de Vries equation, Nonlinear Anal. 31 (1998), no. 5-6, 599–619. MR 1487849, DOI 10.1016/S0362-546X(97)00426-4
- Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 235310, DOI 10.1002/cpa.3160210503
- Wen Xiu Ma, Complexiton solutions to the Korteweg-de Vries equation, Phys. Lett. A 301 (2002), no. 1-2, 35–44. MR 1927047, DOI 10.1016/S0375-9601(02)00971-4
- Franco Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156–1162. MR 488516, DOI 10.1063/1.523777
- V. B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A 166 (1992), no. 3-4, 205–208. MR 1170966, DOI 10.1016/0375-9601(92)90362-P
- V. B. Matveev and M. A. Salle, Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. MR 1146435, DOI 10.1007/978-3-662-00922-2
- Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204. MR 252825, DOI 10.1063/1.1664700
- C. Rasinariu, U. Sukhatme, and Avinash Khare, Negaton and positon solutions of the KdV and mKdV hierarchy, J. Phys. A 29 (1996), no. 8, 1803–1823. MR 1395807, DOI 10.1088/0305-4470/29/8/027
- J. Satsuma, A Wronskian representation of $N$-soliton solutions of nonlinear evolution equations, J. Phys. Soc. Jpn. 46 (1979) 359–360.
- Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65. MR 783348, DOI 10.1007/BF02698802
- James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888, DOI 10.1007/978-94-010-0732-0
- S. Sirianunpiboon, S. D. Howard, and S. K. Roy, A note on the Wronskian form of solutions of the KdV equation, Phys. Lett. A 134 (1988), no. 1, 31–33. MR 972621, DOI 10.1016/0375-9601(88)90541-5
- Yuncheng You, Global dynamics of dissipative generalized Korteweg-de Vries equations, Chinese Ann. Math. Ser. B 17 (1996), no. 4, 389–402. A Chinese summary appears in Chinese Ann. Math. Ser. A 17 (1996), no. 4, 651. MR 1441652
Additional Information
- Wen-Xiu Ma
- Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
- MR Author ID: 247034
- ORCID: 0000-0001-5309-1493
- Email: mawx@math.usf.edu
- Yuncheng You
- Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
- Email: you@math.usf.edu
- Received by editor(s): June 2, 2003
- Published electronically: December 22, 2004
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 1753-1778
- MSC (2000): Primary 35Q53, 37K10; Secondary 35Q51, 37K40
- DOI: https://doi.org/10.1090/S0002-9947-04-03726-2
- MathSciNet review: 2115075