Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Nonlinear evolution of wave packets in magnetic fluids

Authors: S. K. Malik and M. Singh
Journal: Quart. Appl. Math. 48 (1990), 415-431
MSC: Primary 76W05
DOI: https://doi.org/10.1090/qam/1074957
MathSciNet review: MR1074957
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Abstract: The propagation of wave packets on the surface of a magnetic fluid of finite depth is considered in $ \left( {2 + 1} \right)$ dimensions. It is shown that the evolution of the envelope is governed by two coupled partial differential equations with cubic nonlinearity. The stability analysis reveals the existence of more than one region of instability. The enveloping soliton and the ``waveguide'' solutions are derived in the regions of instability wherever they exist in one space dimension. The instability regions are sensitive to the applied magnetic field strength. The evolution of the envelope is governed by a $ \left( {2 + 1} \right)$-dimensional nonlinear equation which leads to a self-focussing singularity. Examined also is the long wave/short wave resonant interaction in magnetic fluids. Our study shows that the application of the magnetic field decreases the region of instability where this resonance occurs.

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  • [1] R. E. Zelazo and J. R. Melcher, Dynamics and stability of ferrofluids: surface interactions, J. Fluid Mech. 39, 1 (1969)
  • [2] R. E. Rosensweig. Ferrohydrodynamics, Cambridge Univ. Press, New York, 1985
  • [3] R. E. Rosensweig, Magnetic fluids, Annal. Rev. Fluid Mech. 19, 437 (1987)
  • [4] R. E. Rosensweig, Fluid dynamics and science of magnetic liquids, Adv. Electronics and Electron Physics, 48, 103 (1979)
  • [5] S. K. Malik and M. Singh, Modulational instability in magnetic fluids, Quart. Appl. Math. 42, 57 (1985) MR 782256
  • [6] S. K. Malik and M. Singh. Nonlinear capillary-gravity waves in magnetic fluids, Quart. Appl. Math. 47, 59-70 (1989) MR 987895
  • [7] M. C. Shen and S. M. Sun, Ray method for surface waves on a ferromagnetic fluid, Wave Motion 9, 99 (1987)
  • [8] A. Davey and K. Stewartson. On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338, 101-110 (1974) MR 0349126
  • [9] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech. 79, 703 (1977) MR 0443555
  • [10] M. J. Ablowitz and H. Segur. On the evolution of packets of water waves, J. Fluid Mech. 92, 691 (1979) MR 544892
  • [11] M. J. Ablowitz and H. Segur, Solitons and the Inverse Spectral Transform, SIAM, 1981 MR 642018
  • [12] S. V. Manakov, Dynamics of classical solitons, Phys. Rep. 35, 1 (1978) MR 481361
  • [13] V. E. Zakharov and S. Synakh, The nature of the self-focussing singularity, Soviet Phys. JETP 41, 465 (1975)
  • [14] V. E. Zakharov, Collapse of Langmuir waves, Soviet Phys. JETP 72, 908 (1972)
  • [15] F. H. Berkshire and J. D. Gibbon, Collapse in the n-dimensional nonlinear Schrödinger equation, Stud. Appl. Math. 69, 229 (1983) MR 721132
  • [16] S. K. Malik and M. Singh, Nonlinear focussing in magnetic fluids, Quart. Appl. Math. 44, 629-637 (1987) MR 872815
  • [17] D. J. Benney, Significant interaction between small and large scale surface waves, Stud. Appl. Math. 55, 93-106 (1976) MR 0452076
  • [18] D. J. Benney, A general theory of interactions between short and long waves, Stud. Appl. Math. 56, 81-94 (1977) MR 0463715
  • [19] T. B. Benjamin and J. E. Feir, The disintegration of wave trains in deep water, J. Fluid Mech. 27, 417 (1967)
  • [20] A. C. Newell, Longwaves-shortwaves, a solvable model, SIAM J. Appl. Math. 35, 650 (1978) MR 512175
  • [21] R. Kant and S. K. Malik, Nonlinear internal resonance in magnetic fluids, J. Mag. Mat. 65, 347 (1987)
  • [22] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focussing and one-dimensinal self-modulating waves in nonlinear media, Soviet Phys. JETP 34, 62 (1972) MR 0406174
  • [23] V. E. Zakharov and A. M. Rubenchik, Instability of waveguides and solutions in nonlinear media, Soviet Phys. JETP 38, 494 (1974)
  • [24] Y. C. Ma, The complete solution of the long wave-short wave resonance equations, Stud. Appl. Math. 59, 201 (1978) MR 0521795
  • [25] B. B. Kadomstev and V. I. Petviashvibi, On the stability of solitary waves in weakly dispersive media, Soviet Phys. Dokl. 15, 539 (1970)
  • [26] S. K. Malik and M. Singh, On long surface waves in magnetic fluids, to appear
  • [27] J. Hammack and H. Segur, The Korteweg-de Vries equation and water waves, J. Fluid Mech. 65, 289 (1974) MR 0366198
  • [28] J. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 3, Oscillatory Waves, J. Fluid Mech. 84, 337 (1978) MR 0502780
  • [29] H. Segur, in Topics in Ocean Physics (A. R. Osborne, ed.), North Holland, New York, 1982 MR 699954
  • [30] M. C. Shen, Solitons on a magnetic fluid, Continuum Mechanics and Its Application (G. A. C. Graham and S. K. Malik, eds.), Hemisphere, New York, 1989 MR 1051688

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DOI: https://doi.org/10.1090/qam/1074957
Article copyright: © Copyright 1990 American Mathematical Society

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