Abstract
The nonlinear Schrödinger equationA t ±iA xx+i‖A‖2 A=0 describes an envelope of periodic waves with slowly varying parameters on water, in plasmas, and in nonlinear optics [1]. This equation can also be applied to steady periodic waves (the wave amplitude and wave number do not depend on time, the variablest andx are replaced by the variables of a horizontal coordinate system on the surface of the fluid [2]). In the present paper the properties of a modified Schrödinger equation involving the third and higher derivatives are studied. Solutions describing transition regions between uniform wave states are obtained numerically. If the structure of the transition region whose extent increases with time is not considered, these solutions may be interpreted as jumps.
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References
R. Dodd, G. Eilbeck, G. Gibbon, and H. Morris,Solitons and Nonlinear Wave Equations [Russian translation], Mir, Moscow (1988).
D. K. P. Yue and C. C. Mei, Forward diffraction of Stokes waves by a thin wedge,”J. Fluid Mech.,99, 33 (1980).
T. R. Akylas and T.-J. Kung, “On nonlinear wave envelopes of permanent form near a caustic,”J. Fluid Mech.,214, 489 (1990).
M. Oikawa, “Nonlinear behavior of capillary-gravity waves near the inflection point of the dispersion curve,”Rep. Res. Inst. Appl. Mech. Japan,38, No. 108, 61 (1991).
A. G. Kulikovskii and V. A. Reutov, “Propagation of nonlinear waves over semi-infinite underwater troughs and crests,”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 53 (1980).
A. G. Kulikovskii, “Stability of uniform states,”Prikl. Mat. Mekh.,30, 148 (1966).
A. V. Marchenko, “Long waves in a shallow fluid beneath an ice sheet,”Prikl. Mat. Mekh.,52, 230 (1988).
A. T. Il'ichev and A. V. Marchenko, “Propagation of long nonlinear waves in a heavy liquid beneath the ice sheet,”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1, 88 (1989).
I. B. Bakholdin, “Averaged equations and discontinuities describing the propagation of Stokes waves with slowly varying parameters,”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 113 (1989).
I. B. Bakholdin, “Discontinuities of the variables characterizing the propagation of solitary waves in a fluid layer,”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3, 87 (1984).
A. A. Barmin and A. G. Kulikovskii, “Shock waves ionizing a gas in an electromagnetic field,”Dokl. Akad. Nauk SSSR,178, 55 (1968).
I. B. Bakholdin, “Three-wave resonance and averaged equations of the interaction of two waves in media described by a cubic Schrödinger equation,”Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 1, 107 (1992).
A. A. Samarskii,Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).
T. R. Akylas, “Unsteady and nonlinear effect near the cusp lines of the Kelvin ship-wave pattern,”J. Fluid Mech.,175, 333 (1980).
A. V. Gurevich and L. P. Pitaevskii, “Nonstationary structure of a collisionless shock wave,”Zh. Eksp. Teor. Fiz.,65, 590 (1973).
Additional information
Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 111–124, July–August, 1994.
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Bakholdin, I.B. Wave jumps in media described by a modified Schrödinger equation. Fluid Dyn 29, 528–539 (1994). https://doi.org/10.1007/BF02319074
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DOI: https://doi.org/10.1007/BF02319074