Nonlinear dispersive instabilities in magnetic fluids

Authors:
S. K. Malik and M. Singh

Journal:
Quart. Appl. Math. **42** (1984), 359-371

MSC:
Primary 76E30; Secondary 35Q20

DOI:
https://doi.org/10.1090/qam/757174

MathSciNet review:
757174

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Abstract | References | Similar Articles | Additional Information

Abstract: An asymptotic nonlinear theory of the two superposed magnetic fluids is presented taking into account the spatial as well as temporal effects. A generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. The various stability criteria are derived from this equation. Obtained also are the bell shaped soliton and the kink solutions.

**[1]**M. D. Cowley and R. E. Rosensweig,*The interfacial instability of a ferromagnetic fluid*, J. Fluid Mech.**30**, 671 (1967)**[2]**R. E. Rosensweig,*Pattern formation in magnetic fluids*, Fourth Internat. Conf. on Synergetics, Ed. by H. Haken, Springer-Verlag, Berlin (1983)**[3]**R. E. Zelazo and J. R. Melcher,*Dynamics and stability of ferro-fluids*, J. Fluid Mech.**39**, 1 (1969)**[4]**M. I. Shlimois and Yu. L. Raikher,*Experimental investigations of magnetic fluids*, I.E.E.E. Trans. Mag.**16**, 237 (1980)**[5]**A. Gailitis,*Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field*, J. Fluid Mech.**82**, 401 (1977)**[6]**E. A. Kuznetsov and M. D. Spektor,*Existence of a hexagonal relief on the surface of a dielectric fluid in an external electric field*, Sov. Phys. JETP**44**, 136 (1976)**[7]**J. P. Brancher,*Waves and instabilities on a plane interface between ferrofluids and nonmagnetic fluids*, Thermo-mechanics of Magnetic Fluids, Ed. by B. Berkovsky, Hemisphere Publishing Corp. (1977)**[8]**E. Twombly and J. W. Thomas,*Theory of non-linear waves on the surface of a magnetic fluid*, I.E.E.E. Trans. on Magnetics**16**, 214 (1980)**[9]**J. T. Stuart,*Bifurcation theory in non-linear hydrodynamic stability*, Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976) Academic Press, New York, 1977, pp. 127–147. Publ. Math. Res. Center, No. 38. MR**0495631****[10]**A. C. Newell,*Envelope equation*, Lect. Appl. Math.**15**, 157 (1974)**[11]**W. Eckhaus,*Studies in nonlinear stability theory*, Springer (1965)**[12]**C. G. Lang and A. C. Newell,*The post-buckling problem for thin elastic shells*, SIAM J. Appl. Math.**21**, 605 (1971)**[13]**J. Pedlosky,*The finite Amplitude Dynamics of Baroclinic Waves*, Application of Bifurcation Theory, Ed. by P. H. Rabinowitz, New York, Academic Press (1977)**[14]**M. A. Weissman,*Nonlinear wave packets in the kelvin-helmholtz instability*, Phil. Trans. Roy. Soc. London**290**, 58 (1979)**[15]**S. K. Malik and M. Singh,*Nonlinear instability in superposed magnetic fluids*, Third Internat. Conf. on Magnetic Fluids (1983)**[16]**G. B. Whitham,*Linear and nonlinear waves*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0483954****[17]**V. G. Makhan′kov,*Dynamics of classical solitons (in nonintegrable systems)*, Phys. Rep.**35**(1978), no. 1, 1–128. MR**481361**, https://doi.org/10.1016/0370-1573(78)90074-1**[18]**V. M. Zaitsev and M. I. Shliomis,*The nature of the instability of the interface between two liquids in a constant field*, Dokl. Akad. Nauk SSR**188**, 1261 (1969)

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DOI:
https://doi.org/10.1090/qam/757174

Article copyright:
© Copyright 1984
American Mathematical Society