By A. E. H. LOVE

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A conﬁrmation of this scenario was obtained by means of numerical experiments. Here we describe relatively simple experiments based on the numerical solutions of the system (92), (93), (63) with simpliﬁcations (95) and (96). Of course, it is a model system based on the assumption of the smallness of parameter µ which is correct only for suﬃciently broad beams. But the NLS equation, predicting self-focusing, must also be considered as a model equation formally valid at small µ . [In fact, it is less accurate than the system (92), (93)].

I. Karpman Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel Abstract. A theory of envelope whistler solitons beyond the approximation based on the nonlinear Schr¨ odinger (NLS) equation is developed. It is shown that such solitons must emanate radiation due to the continuos transformation of trapped whistler modes into other modes that cannot be trapped in the duct, produced by the soliton (such modes are not described by the NLS equation). An equation governing the decrease of soliton amplitude due to the loss of trapped radiation is derived.

1982, The Alfv´en Wave, (Tech. Inf. Center, U. S. Department of Energy, Washington, D. ). 13. Kennel C. , 1988, Phys. Fluids, 31, 1949. 14. D. Leneman, W. Gekelman, & J. Maggs, 1999, Phys. Rev. Lett, 82, 2673. 15. , 1990, Space Sci. , 52, 33. 16. , 1986, J. , 36, 1. 17. Lorenz E. , 1963, J. Atmos. , 20, 130. 18. M¨ akel¨ a J. , 1998, J. Geophys. , 103, 9391. Nonlinear Dispersive Alfv´en Waves 29 19. Mikhailovskii A. , Petviashvili V. , & Friedman A. , 1977, JETP Letters, 24, 43. 20. Mikhailovskii A.