The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation: IV. Time dependent coefficients
Author:
J. A. Leach
Journal:
Quart. Appl. Math. 76 (2018), 361-382
MSC (2010):
Primary 35Q53
DOI:
https://doi.org/10.1090/qam/1481
Published electronically:
September 20, 2017
MathSciNet review:
3769899
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Abstract: In this paper, we consider an initial-value problem for the Korteweg-de Vries equation with time dependent coefficients. The normalized variable coefficient Korteweg-de Vries equation considered is given by \begin{equation*} u_{t}+ \Phi (t) u u_{x}+ \Psi (t) u_{xxx}=0, \quad -\infty <x<\infty , \quad t>0, \end{equation*} where $x$ and $t$ represent dimensionless distance and time respectively, whilst $\Phi (t)$, $\Psi (t)$ are given functions of $t (>0)$. In particular, we consider the case when the initial data has a discontinuous expansive step, where $u(x,0)=u_{+}$ for $x \ge 0$ and $u(x,0)=u_{-}$ for $x<0$. We focus attention on the case when $\Phi (t)=t^{\delta }$ (with $\delta >-\frac {2}{3}$) and $\Psi (t)=1$. The constant states $u_{+}$, $u_{-}$ ($<u_{+}$) and $\delta$ are problem parameters. The method of matched asymptotic coordinate expansions is used to obtain the large-$t$ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in $x \ge \frac {u_{-} }{(\delta +1)}t^{(\delta +1)}$ as $t \to \infty$, while the solution is oscillatory in $x<\frac {u_{-}}{(\delta +1)}t^{(\delta +1)}$ as $t \to \infty$. We conclude with a brief discussion of the structure of the large-$t$ solution of the initial-value problem when the initial data is step-like being continuous with algebraic decay as $|x| \to \infty$, with $u(x,t) \to u_{+}$ as $x \to \infty$ and $u(x,t) \to u_{-} (<u_{+})$ as $x \to -\infty$.
References
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- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 404890, DOI https://doi.org/10.1137/1018076
- N. Nirmala, M. J. Vedan, and B. V. Baby, Auto-Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg-de Vries equation. I, J. Math. Phys. 27 (1986), no. 11, 2640–2643. MR 861323, DOI https://doi.org/10.1063/1.527282
- Lord Rayleigh, On waves, Philos. Mag. 1 (1876), 257–279.
- Alwyn Scott (ed.), Encyclopedia of nonlinear science, Routledge, New York, 2005. MR 2265413
- H. Segur and M. J. Ablowitz, Asymptotic solutions of non-linear evolution equations and a Painleve transcendent, Physica D (1981), 165–184.
- M. Vlieg-Hulstman and W. D. Halford, Exact solutions to KdV equations with variable coefficients and/or nonuniformities, Comput. Math. Appl. 29 (1995), no. 1, 39–47. MR 1308136, DOI https://doi.org/10.1016/0898-1221%2894%2900205-Y
- H. Washimi and T. Taniuti, Propergation of ion-accoustic solitary waves of small amplitude, Phys. Rev. Lett. 17 (1966), 996–998.
- L. van Wijngaarden, On the equation of motion for mixtures of liquid and gas bubbles, J. Fluid Mech. 33 (1968), 465-474.
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References
- M. J. Ablowitz and H. Segur, Asymptotic solutions of the Korteweg-deVries equation, Studies in Appl. Math. 57 (1976/77), no. 1, 13–44. MR 0481656
- Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Third printing, with corrections. National Bureau of Standards Applied Mathematics Series, vol. 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1965. MR 0177136
- J. Boussinesq, Théorie de l’intumescence liquid appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, Comptes Rendus Acad. Sci. 72 (1871) 755-759.
- John P. Boyd, Weakly nonlocal solitary waves and beyond-all-orders asymptotics, Mathematics and its Applications, vol. 442, Kluwer Academic Publishers, Dordrecht, 1998. Generalized solitons and hyperasymptotic perturbation theory. MR 1636975
- Radu C. Cascaval, Variable coefficient KdV equations and waves in elastic tubes, Evolution equations, Lecture Notes in Pure and Appl. Math., vol. 234, Dekker, New York, 2003, pp. 57–69. MR 2073735
- G. A. El and R. H. J. Grimshaw, Generation of undular bores in the shelves of slowly-varying solitary waves, Chaos 12 (2002), no. 4, 1015–1026. MR 1946775, DOI https://doi.org/10.1063/1.1507381
- F. B. Ali Sulaiman, J. A. Leach, and D. J. Needham, The large-time solution of Burgers’ equation with time-dependent coefficients. II. Algebraic coefficients, Stud. Appl. Math. 137 (2016), no. 3, 273–305. MR 3564301, DOI https://doi.org/10.1111/sapm.12130
- A. P. Fordy, Soliton theory: a brief synopsis, Soliton theory: a survey of results, Nonlinear Sci. Theory Appl., Manchester Univ. Press, Manchester, 1990, pp. 3–22. MR 1090584
- Zun Tao Fu, Shi Da Liu, Shi Kuo Liu, and Qiang Zhao, New exact solutions to KdV equations with variable coefficients or forcing, Appl. Math. Mech. 25 (2004), no. 1, 67–73 (Chinese, with English and Chinese summaries); English transl., Appl. Math. Mech. (English Ed.) 25 (2004), no. 1, 73–79. MR 2041342, DOI https://doi.org/10.1007/BF02437295
- Katrin Grunert and Gerald Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom. 12 (2009), no. 3, 287–324. MR 2525595, DOI https://doi.org/10.1007/s11040-009-9062-2
- R. Grimshaw, Slowly varying solitary waves. I. Korteweg - de Vries equation, Proc. Roy. Soc. London Ser. A 368 (1979), no. 1734, 359–375. MR 551468, DOI https://doi.org/10.1098/rspa.1979.0135
- Roger Grimshaw, Efim Pelinovsky, and Tatjana Talipova, Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity, Phys. D 132 (1999), no. 1-2, 40–62. MR 1705698, DOI https://doi.org/10.1016/S0167-2789%2899%2900045-7
- S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), no. 1, 31–51. MR 555581, DOI https://doi.org/10.1007/BF00283254
- Karl R. Helfrich and W. Kendall Melville, Long nonlinear internal waves, Annual review of fluid mechanics. Vol. 38, Annu. Rev. Fluid Mech., vol. 38, Annual Reviews, Palo Alto, CA, 2006, pp. 395–425. MR 2206980, DOI https://doi.org/10.1146/annurev.fluid.38.050304.092129
- E.M. de Jager, On the origin of the Korteweg-de Vries equation, ArXiv: math.HO/0602661 (2006)
- A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Rev. 14 (1972), 582–643. MR 0334675, DOI https://doi.org/10.1137/1014101
- R. S. Johnson, On the development of a solitary wave moving over an uneven bottom, Proc. Cambridge Philos. Soc. 73 (1973), 183–203. MR 0312817
- R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555
- R.S. Johnson, Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf), J. Fluid Mech. 54 (1972) 81-91.
- Nalini Joshi, Painlevé property of general variable-coefficient versions of the Korteweg-de Vries and nonlinear Schrödinger equations, Phys. Lett. A 125 (1987), no. 9, 456–460. MR 917430, DOI https://doi.org/10.1016/0375-9601%2887%2990184-8
- D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. (5) 39 (1895), no. 240, 422–443. MR 3363408
- J. A. Leach, The large-time solution of Burgers’ equation with time-dependent coefficients. I. The coefficients are exponential functions, Stud. Appl. Math. 136 (2016), no. 2, 163–188. MR 3457798, DOI https://doi.org/10.1111/sapm.12098
- J. A. Leach and D. J. Needham, Matched asymptotic expansions in reaction-diffusion theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2004. MR 2013330
- J. A. Leach and D. J. Needham, The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation. I. Initial data has a discontinuous expansive step, Nonlinearity 21 (2008), no. 10, 2391–2408. MR 2439485, DOI https://doi.org/10.1088/0951-7715/21/10/010
- J. A. Leach and D. J. Needham, The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation. II. Initial data has a discontinuous compressive step, Mathematika 60 (2014), no. 2, 391–414. MR 3229496, DOI https://doi.org/10.1112/S0025579313000284
- W. Lick, Nonlinear wave propergation in fluids, Annu. Rev. Fluid Mech. 2 (1970), 113–134.
- S. Maxon and J. Viecelli, Cylindrical solitons, Phys. Fluids 17 (1974), 1614-1616.
- S. Maxon and J. Viecelli, Spherical solitons, Phys. Rev. Lett. 32 (1974), 4-6.
- C.C. Mei. The Applied Dynamics of Ocean Surface Waves. World Scientific, Singapore (1989)
- John W. Miles, Solitary waves, Annual review of fluid mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 1980, pp. 11–43. MR 565388
- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 0404890, DOI https://doi.org/10.1137/1018076
- N. Nirmala, M. J. Vedan, and B. V. Baby, Auto-Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg-de Vries equation. I, J. Math. Phys. 27 (1986), no. 11, 2640–2643. MR 861323, DOI https://doi.org/10.1063/1.527282
- Lord Rayleigh, On waves, Philos. Mag. 1 (1876), 257–279.
- Alwyn Scott (ed.), Encyclopedia of nonlinear science, Routledge, New York, 2005. MR 2265413
- H. Segur and M. J. Ablowitz, Asymptotic solutions of non-linear evolution equations and a Painleve transcendent, Physica D (1981), 165–184.
- M. Vlieg-Hulstman and W. D. Halford, Exact solutions to KdV equations with variable coefficients and/or nonuniformities, Comput. Math. Appl. 29 (1995), no. 1, 39–47. MR 1308136, DOI https://doi.org/10.1016/0898-1221%2894%2900205-Y
- H. Washimi and T. Taniuti, Propergation of ion-accoustic solitary waves of small amplitude, Phys. Rev. Lett. 17 (1966), 996–998.
- L. van Wijngaarden, On the equation of motion for mixtures of liquid and gas bubbles, J. Fluid Mech. 33 (1968), 465-474.
- N. J. Zabusky, Solitons and energy transport in nonlinear latticies, Comput. Phys. Commun. 5 (1973), 1–10.
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J. A. Leach
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, U.K.
MR Author ID:
323003
Received by editor(s):
March 27, 2017
Received by editor(s) in revised form:
August 2, 2017
Published electronically:
September 20, 2017
Article copyright:
© Copyright 2017
Brown University