A global theory of internal solitary waves in two-fluid systems

Authors:
C. J. Amick and R. E. L. Turner

Journal:
Trans. Amer. Math. Soc. **298** (1986), 431-484

MSC:
Primary 35Q99; Secondary 76C10

DOI:
https://doi.org/10.1090/S0002-9947-1986-0860375-3

MathSciNet review:
860375

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity . It is assumed that a fluid of constant density lies above a fluid of constant density and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair , where the constant , being the wave speed, and where is the height at a horizontal position of the streamline which has height at . It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for is an unbounded connected set in . Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.

**[1]**H. W. Alt, L. A. Caffarelli, and A. Friedman,*Variational problems with two phases and their free boundaries*, Trans. Amer. Math. Soc.**282**(1984), 431-461. MR**732100 (85h:49014)****[2]**-,*Jets with two fluids*. I.*One free boundary*, Indiana Univ. Math. J.**33**(1984), 213-248. MR**733897 (86a:35144)****[3]**S. Agmon, A. Douglas, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions*. II, Comm. Pure Appl. Math.**17**(1964), 35-92. MR**0162050 (28:5252)****[4]**C. J. Amick,*Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids*, Ann. Scuola Norm. Sup. Pisa Sér. (4)**11**(1984), 441-499. MR**785621 (86i:35042)****[5]**C. J. Amick and J. F. Toland,*On solitary water-waves of finite amplitude*, Arch. Rational Mech. Anal.**76**(1981), 9-95. MR**629699 (83b:76017)****[6]**J. T. Beale,*The existence of solitary water waves*, Comm. Pure Appl. Math.**30**(1977), 373-389. MR**0445136 (56:3480)****[7]**H. Beirão da Veiga, R. Serapioni, and A. Valli,*On the motion of nonhomogeneous fluids in the presence of diffusion*, J. Math. Anal. Appl.**85**(1982), 179-191. MR**647566 (84a:76042)****[8]**T. B. Benjamin,*Internal waves of finite amplitude and permanent form*, J. Fluid Mech.**25**(1966), 241-270.**[9]**-,*A unified theory of conjugate flows*, Philos. Trans. Roy. Soc. London A**269**(1971), 587-643. MR**0446075 (56:4407)****[10]**J. L. Bona, D. K. Bose, and R. E. L. Turner,*Finite amplitude steady waves in stratified fluids*, J. Math. Pure Appl.**62**(1983), 389-439. MR**735931 (85e:76056)****[11]**A. D. Frank-Kamenetskii,*Diffusion and heat transfer in chemical kinetics*, 2nd ed., Plenum, New York-London, 1969 [translated from Russian].**[12]**K. O. Friedrichs and D. H. Hyers,*The existence of solitary waves*, Comm. Pure Appl. Math.**7**(1954), 517-550. MR**0065317 (16:413f)****[13]**D. Gilbarg and N. S. Trudinger,*Elliptic partial differential equations*, Academic Press, New York, 1968.**[14]**A. V. Kazhikhov and Sh. Smagulov,*The correctness of boundary value problems in a diffusion model of inhomogeneous liquid*, Soviet Phys. Dokl.**22**(1977), 249-250 (Dokl. Akad. Nauk SSSR**234**(1977), 330-332 [in Russian]). MR**0446059 (56:4391)****[15]**D. Kinderlehrer, L. Nirenberg, and J. Spruck,*Regularity in elliptic free boundary problems*. I, J. Analyse Math.**34**(1978), 86-119. MR**531272 (83d:35060)****[16]**R. R. Long,*Solitary waves in the one- and two-fluid systems*, Tellus**8**(1956), 460-471.**[17]**D. I. Meiron and P. G. Saffman,*Overhanging interfacial gravity waves of large amplitude*, J. Fluid Mech.**129**(1983), 213-218.**[18]**N. G. Meyers,*An*-*estimate for the gradient of solutions of second order elliptic divergence equations*, Ann. Scuola Norm. Sup. Pisa Sér. (3)**17**(1963), 189-205. MR**0159110 (28:2328)****[19]**R. M. Miura,*The Korteweg-de Vries equation*:*a survey of results*, SIAM Rev.**18**(1976), 412-459. MR**0404890 (53:8689)****[20]**C. B. Morrey, Jr.,*On the analyticity of the solutions of analytic nonlinear elliptic systems of partial differential equations*. I, Amer. J. Math.**80**(1958), 198-218; II 219-234. MR**0106336 (21:5070)****[21]**-,*Multiple integrals in the calculus of variations*, Springer-Verlag, New York-Berlin, 1966. MR**0202511 (34:2380)****[22]**L. Nirenberg,*On elliptic partial differential equations*, Ann. Sculoa Norm. Sup. Pisa Sér. (3)**13**(1959), 115-162. MR**0109940 (22:823)****[23]**A. S. Peters and J. J. Stoker,*Solitary waves in liquids having non-constant density*, Comm. Pure Appl. Math.**8**(1960), 115-164. MR**0112445 (22:3296)****[24]**J. S. Russell,*Report on waves*, "Rep. 14th Meeting of the British Association for the Advancement of Science", p. 311, John Murray, London, 1844.**[25]**A. M. Ter-Krikorov,*Théorie exacte des ondes longues stationnaires dans un liquide hétérogène*, J. Mécanique**2**(1963), 351-376. MR**0160400 (28:3613)****[26]**R. E. L. Turner,*Internal waves in fluids with rapidly varying density*, Ann. Scuola Norm. Sup. Pisa Sér. (4)**8**(1981), 513-573. MR**656000 (83j:76027)****[27]**-,*A variational approach to surface solitary waves*, J. Differential Equations**55**(1984), 401-438. MR**766131 (86a:76012)****[28]**R. E. L. Turner and J.-M. Vanden-Broeck,*On the limiting configuration of interfacial gravity waves*, Mathematics Research Center Technical Summary Report #2800, University of Wisconsin-Madison, Madison, Wisconsin, Phys. Fluids**29**(1986), 372-375. MR**828169 (87j:76033)****[29]**-,*On solitary waves in two-fluid systems*(to appear).**[30]**L. R. Walker,*Interfacial solitary waves in a two-fluid medium*, Phys. Fluids**16**(1973), 1796-1804.**[31]**G. T. Whyburn,*Topological analysis*, Princeton Univ. Press, Princeton, N.J., 1958. MR**0099642 (20:6081)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35Q99,
76C10

Retrieve articles in all journals with MSC: 35Q99, 76C10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0860375-3

Keywords:
Internal wave,
solitary wave,
internal bore,
bifurcation

Article copyright:
© Copyright 1986
American Mathematical Society