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Dromions and a boundary value problem for the Davey-Stewartson 1 equation. (English) Zbl 0707.35144

Localized solitons in the plane have been a hot topic since their discovery in 1988; see here M. Boiti, J. Leon, L. Martina, and F. Pempinelli (BLMP), Phys. Lett. A 132, 432-439 (1988), and Nonlinear evolution equations..., Manchester Univ. Press, 1990, pp. 249-262 (Proc. Workshop, Como, 1988).
The localized traveling solutions (dromions) studied here are more general; for special parameter values they reduce to the localized solitons of BLMP. We extract instead from the abstract: We solve an initial-boundary value problem for the Devey-Stewartson (DS) I equation, which is a 2 dimensional generalization of the nonlinear Schrödinger equation. This equation, which describes the interaction of a surface wave envelope of amplitude q(x,y,t) with the mean flow, arises in a whole range of physical problems. We find that the energy from the mean flow can be transferred to the surface envelope and create focusing effects. Indeed, for generic non zero boundary conditions on the mean flow, an arbitrary initial envelope q(x,y,t) will form a number of 2 dimensional, exponentially decaying in both x and y, localized structures.
Furthermore, in contrast to the one dimensional solitons, these solutions do not preserve their form upon interaction and hence can exchange energy. These coherent structures can be driven everywhere in the plane by choosing a suitable motion for the boundaries. We call these novel localized coherent structures dromions.
Reviewer: R.Carroll

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
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[1] Davey, A.; Stewartson, K., (Proc. R. Soc. London Ser. A, 338 (1974)), 101 · Zbl 0282.76008
[2] Calogero, F.; Maccari, A., (Sabatier, P. C., Inverse Problems; An Interdisciplinary Study (1987), Academic Press: Academic Press New York)
[3] Benney, D. J.; Roskes, G. J., Stud. Appl. Math., 48, 377 (1969) · Zbl 0216.52904
[4] Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076
[5] Cornille, H., J. Math. Phys., 20, 199 (1979) · Zbl 0425.35087
[6] Fokas, A. S., Phys. Rev. Lett., 51, 3-6 (1983)
[7] Kaup, D. J., Physica D, 1, 45 (1980) · Zbl 1194.35372
[8] Schultz, C.; Ablowitz, M. J., Trace formula for DS1 in t̄6 limit case, (INS, 91 (1988), Clarkson University), preprint
[9] Boiti, M.; Leon, J.; Martina, L.; Pempinelli, F., Phys. Lett. A, 132, 432-439 (1988)
[10] Fokas, A. S.; Santini, P. M., Phys. Rev. Lett., 63, 1329 (1989)
[11] Kaup, D. J., J. Math. Phys., 22, 1176 (1981) · Zbl 0467.35070
[12] Gardner, C. S.; Greene, J. M.; Krushkal, M. D.; Miura, R. M., Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520
[13] Santini, P. M., Energy exchange of interacting coherent structures in multidimensions, Physica D, 41, 26-54 (1990) · Zbl 0696.35180
[14] Fokas, A. S., Physica D, 35, 167-185 (1989) · Zbl 0679.35076
[15] Calogero, F.; Degasperis, A., (Spectral Transform and Solitons, I (1982), North-Holland: North-Holland Amsterdam) · Zbl 0596.35009
[16] Fokas, A. S.; Ablowitz, M. J., Stud. Appl. Math., 69, 211-228 (1983) · Zbl 0528.35079
[17] Manakov, S. V., Physica D, 3, 420 (1981) · Zbl 1194.35507
[18] Fokas, A. S.; Ablowitz, M. J., Phys. Rev. Lett., 47, 1096 (1981)
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