Camassa–Holm equation
In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation
\[ u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. \, \]
The equation was introduced by Camassa and Holm[1] as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons.
In the special case that κ is equal to zero, the Camassa–Holm equation has peakon solutions: solitons with a sharp peak, so with a discontinuity at the peak in the wave slope.
Contents
Relation to waves in shallow water
The Camassa–Holm equation can be written as the system of equations:[2]
\[ \begin{align} u_t + u u_x + p_x &= 0, \\ p - p_{xx} &= 2 \kappa u + u^2 + \frac{1}{2} \left( u_x \right)^2, \end{align} \]
with p the (dimensionless) pressure or surface elevation. This shows that the Camassa–Holm equation is a model for shallow water waves with non-hydrostatic pressure and a water layer on a horizontal bed.
The linear dispersion characteristics of the Camassa–Holm equation are:
\[\omega = 2\kappa \frac{k}{1+k^2},\]
with ω the angular frequency and k the wavenumber. Not surprisingly, this is of similar form as the one for the Korteweg–de Vries equation, provided κ is unequal to zero. For κ equal to zero, the Camassa–Holm equation has no frequency dispersion — moreover, the linear phase speed is zero for this case. As a result, κ is the phase speed for the long-wave limit of k approaching zero, and the Camassa–Holm equation is (if κ is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation.
Hamiltonian structure
Introducing the momentum m as
\[m = u - u_{xx} + \kappa, \,\]
then two compatible Hamiltonian descriptions of the Camassa–Holm equation are:[3]
\[ \begin{align} m_t &= -\mathcal{D}_1 \frac{\delta \mathcal{H}_1}{\delta m} & & \text{ with }& \mathcal{D}_1 &= m \frac{\partial}{\partial x} + \frac{\partial}{\partial x} m & \text{ and } \mathcal{H}_1 &= \frac{1}{2} \int u^2 + \left(u_x\right)^2\; \text{d}x, \\ m_t &= -\mathcal{D}_2 \frac{\delta \mathcal{H}_2}{\delta m} & & \text{ with }& \mathcal{D}_2 &= \frac{\partial}{\partial x} + \frac{\partial^3}{\partial x^3} & \text{ and } \mathcal{H}_2 &= \frac{1}{2} \int u^3 + u \left(u_{xx}\right)^2 - \kappa u^2\; \text{d}x. \end{align} \]
Integrability
The Camassa–Holm equation is an integrable system. Integrability means that there is a change of variables (action-angle variables) such that the evolution equation in the new variables is equivalent to a linear flow at constant speed. This change of variables is achieved by studying an associated isospectral/scattering problem, and is reminiscent of the fact that integrable classical Hamiltonian systems are equivalent to linear flows at constant speed on tori. The Camassa–Holm equation is integrable provided that the momentum
\[ m= u-u_{xx}+ \kappa \, \]
is positive — see [4] and [5] for a detailed description of the spectrum associated to the isospectral problem,[4] for the inverse spectral problem in the case of spatially periodic smooth solutions, and [6] for the inverse scattering approach in the case of smooth solutions that decay at infinity.
Exact solutions
Traveling waves are solutions of the form
\[ u(t,x)=f(x-ct) \, \]
representing waves of permanent shape f that propagate at constant speed c. These waves are called solitary waves if they are localized disturbances, that is, if the wave profile f decays at infinity. If the solitary waves retain their shape and speed after interacting with other waves of the same type, we say that the solitary waves are solitons. There is a close connection between integrability and solitons. [7] In the limiting case when κ = 0 the solitons become peaked (shaped like the graph of the function f(x) = e-|x|), and they are then called peakons. It is possible to provide explicit formulas for the peakon interactions, visualizing thus the fact that they are solitons.[8] For the smooth solitons the soliton interactions are less elegant.[9] This is due in part to the fact that, unlike the peakons, the smooth solitons are relatively easy to describe qualitatively — they are smooth, decaying exponentially fast at infinity, symmetric with respect to the crest, and with two inflection points[10] — but explicit formulas are not available. Notice also that the solitary waves are orbitally stable i.e. their shape is stable under small perturbations, both for the smooth solitons[10] and for the peakons.[11]
Wave breaking
The Camassa–Holm equation models breaking waves: a smooth initial profile with sufficient decay at infinity develops into either a wave that exists for all times or into a breaking wave (wave breaking[12] being characterized by the fact that the solution remains bounded but its slope becomes unbounded in finite time). The fact that the equations admits solutions of this type was discovered by Camassa and Holm[1] and these considerations were subsequently put on a firm mathematical basis.[13] It is known that the only way singularities can occur in solutions is in the form of breaking waves.[14] Moreover, from the knowledge of a smooth initial profile it is possible to predict (via a necessary and sufficient condition) whether wave breaking occurs or not.[15] As for the continuation of solutions after wave breaking, two scenarios are possible: the conservative case[16] and the dissipative case[17] (with the first characterized by conservation of the energy, while the dissipative scenario accounts for loss of energy due to breaking).
Long-time asymptotics
It can be shown that for sufficiently fast decaying smooth initial conditions with positive momentum splits into a finite number and solitons plus a decaying dispersive part. More precisely, one can show the following for \(\kappa>0\):[18] Abbreviate \(c =x / (\kappa t)\). In the soliton region \(c>2\) the solutions splits into a finite linear combination solitons. In the region \(0<c<2\) the solution is asymptotically given by a modulated sine function whose amplitude decays like \(t^{-1/2}\). In the region \(-1/4<c<0\) the solution is asymptotically given by a sum of two modulated sine function as in the previous case. In the region \(c<-1/4\) the solution decays rapidly. In the case \(\kappa=0\) it is conjectured[19] that the solution splits into an infinite linear combination of peakons.
See also
Notes
- ↑ 1.0 1.1 Camassa & Holm 1993
- ↑ Loubet 2005
- ↑ Boldea 1995
- ↑ 4.0 4.1 Constantin & McKean 1999
- ↑ Constantin 2001
- ↑ Constantin, Gerdjikov & Ivanov 2006
- ↑ Drazin, P. G.; Johnson, R. S. (1989), Solitons: an introduction, Cambridge University Press, Cambridge
- ↑ Beals, Sattinger & Szmigielski 1999
- ↑ Parker 2005b
- ↑ 10.0 10.1 Constantin & Strauss 2002
- ↑ Constantin & Strauss 2000
- ↑ Whitham, G. B. (1974), Linear and nonlinear waves, Wiley Interscience, New York–London–Sydney
- ↑ Constantin & Escher 1998b
- ↑ Constantin 2000, Constantin & Escher 2000
- ↑ McKean 2004
- ↑ Bressan & Constantin 2007a
- ↑ Bressan & Constantin 2007b
- ↑ Boutet de Monvel, Kostenko, Shepelsky & Teschl 2009
- ↑ McKean 2003a
References
- Beals, Richard; Sattinger, David H.; Szmigielski, Jacek (1999), "Multi-peakons and a theorem of Stieltjes", Inverse Problems 15 (1): L1–L4, arXiv:solv-int/9903011, Bibcode 1999InvPr..15L...1B, doi:10.1088/0266-5611/15/1/001
- Boldea, Costin-Radu (1995), "A generalization for peakon's solitary wave and Camassa–Holm equation", General Mathematics 5 (1–4): 33–42, http://www.emis.de/journals/GM/vol5/bold.html
- Boutet de Monvel, Anne; Kostenko, Aleksey; Shepelsky, Dmitry; Teschl, Gerald (2009), "Long-Time Asymptotics for the Camassa–Holm Equation", SIAM J. Math. Anal. 41 (4): 1559–1588, arXiv:0902.0391, doi:10.1137/090748500
- Bressan, Alberto; Constantin, Adrian (2007a), "Global conservative solutions of the Camassa–Holm equation", Arch. Ration. Mech. Anal. 183 (2): 215–239, Bibcode 2007ArRMA.183..215B, doi:10.1007/s00205-006-0010-z, http://www.math.ntnu.no/conservation/2005/016.html
- Bressan, Alberto; Constantin, Adrian (2007b), "Global dissipative solutions of the Camassa–Holm equation", Anal. Appl. 5: 1–27, doi:10.1142/S0219530507000857, http://www.math.ntnu.no/conservation/2006/023.html
- Camassa, Roberto; Holm, Darryl D. (1993), "An integrable shallow water equation with peaked solitons", Phys. Rev. Lett. 71 (11): 1661–1664, arXiv:patt-sol/9305002, Bibcode 1993PhRvL..71.1661C, doi:10.1103/PhysRevLett.71.1661
- Constantin, Adrian (2000), "Existence of permanent and breaking waves for a shallow water equation: a geometric approach", Annales de l'Institut Fourier 50 (2): 321–362, http://www.numdam.org/item?id=AIF_2000__50_2_321_0
- Constantin, Adrian (2001), "On the scattering problem for the Camassa–Holm equation", R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2008): 953–970, Bibcode 2001RSPSA.457..953C, doi:10.1098/rspa.2000.0701
- Constantin, Adrian; Escher, Joachim (1998b), "Wave breaking for nonlinear nonlocal shallow water equations", Acta Math. 181 (2): 229–243, doi:10.1007/BF02392586
- Constantin, Adrian; Escher, Joachim (2000), "On the blow-up rate and the blow-up set of breaking waves for a shallow water equation", Math. Z. 233 (1): 75–91, doi:10.1007/PL00004793
- Constantin, Adrian; McKean, Henry P. (1999), "A shallow water equation on the circle", Commun. Pure Appl. Math. 52 (8): 949–982, doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
- Constantin, Adrian; Strauss, Walter A. (2000), "Stability of peakons", Comm. Pure Appl. Math. 53 (5): 603–610, doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
- Constantin, Adrian; Strauss, Walter A. (2002), "Stability of the Camassa–Holm solitons", J. Nonlinear Sci. 12 (4): 415–422, Bibcode 2002JNS....12..415C, doi:10.1007/s00332-002-0517-x
- Constantin, Adrian; Gerdjikov, Vladimir S.; Ivanov, Rossen I. (2006), "Inverse scattering transform for the Camassa–Holm equation", Inverse Problems 22 (6): 2197–2207, arXiv:nlin/0603019, Bibcode 2006InvPr..22.2197C, doi:10.1088/0266-5611/22/6/017
- Loubet, Enrique (2005), "About the explicit characterization of Hamiltonians of the Camassa–Holm hierarchy", J. Nonlinear Math. Phys. 12 (1): 135–143, Bibcode 2005JNMP...12..135L, doi:10.2991/jnmp.2005.12.1.11
- McKean, Henry P. (2003a), "Fredholm determinants and the Camassa–Holm hierarchy", Comm. Pure Appl. Math. 56 (5): 638–680, doi:10.1002/cpa.10069
- McKean, Henry P. (2004), "Breakdown of the Camassa–Holm equation", Comm. Pure Appl. Math. 57 (3): 416–418, doi:10.1002/cpa.20003
- Parker, Allen (2005b), "On the Camassa–Holm equation and a direct method of solution. III. N-soliton solutions", Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2064): 3893–3911, Bibcode 2005RSPSA.461.3893P, doi:10.1098/rspa.2005.1537
Further reading
- Peakon solutions
- Beals, Richard; Sattinger, David H.; Szmigielski, Jacek (2000), "Multipeakons and the classical moment problem", Adv. Math. 154 (2): 229–257, arXiv:solv-int/9906001, doi:10.1006/aima.1999.1883
- Water wave theory
- Constantin, Adrian; Lannes, David (2007), "The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations", Preprint arXiv:0709.0905v1 [math.AP], arXiv:0709.0905, Bibcode 2009ArRMA.192..165C, doi:10.1007/s00205-008-0128-2
- Johnson, Robin S. (2003b), "The classical problem of water waves: a reservoir of integrable and nearly-integrable equations", J. Nonlinear Math. Phys. 10 (suppl. 1): 72–92, Bibcode 2003JNMP...10S..72J, doi:10.2991/jnmp.2003.10.s1.6
- Existence, uniqueness, wellposedness, stability, propagation speed, etc.
- Bressan, Alberto; Constantin, Adrian (2007a), "Global conservative solutions of the Camassa–Holm equation", Arch. Ration. Mech. Anal. 183 (2): 215–239, Bibcode 2007ArRMA.183..215B, doi:10.1007/s00205-006-0010-z, http://www.math.ntnu.no/conservation/2005/016.html
- Constantin, Adrian; Strauss, Walter A. (2000), "Stability of peakons", Comm. Pure Appl. Math. 53 (5): 603–610, doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
- Holden, Helge; Raynaud, Xavier (2007a), "Global conservative multipeakon solutions of the Camassa–Holm equation", J. Hyperbolic Differ. Equ. 4 (1): 39–64, doi:10.1142/S0219891607001045, http://www.math.ntnu.no/conservation/2006/011.html
- McKean, Henry P. (2004), "Breakdown of the Camassa–Holm equation", Comm. Pure Appl. Math. 57 (3): 416–418, doi:10.1002/cpa.20003
- Travelling waves
- Lenells, Jonatan (2005c), "Traveling wave solutions of the Camassa–Holm equation", J. Differential Equations 217 (2): 393–430, doi:10.1016/j.jde.2004.09.007
- Integrability structure (symmetries, hierarchy of soliton equations, conservations laws) and differential-geometric formulation
- Fuchssteiner, Benno (1996), "Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation", Physica D 95 (3–4): 229–243, doi:10.1016/0167-2789(96)00048-6
- Lenells, Jonatan (2005a), "Conservation laws of the Camassa–Holm equation", J. Phys. A 38 (4): 869–880, Bibcode 2005JPhA...38..869L, doi:10.1088/0305-4470/38/4/007
- McKean, Henry P. (2003b), "The Liouville correspondence between the Korteweg–de Vries and the Camassa–Holm hierarchies", Comm. Pure Appl. Math. 56 (7): 998–1015, doi:10.1002/cpa.10083
- Misiołek, Gerard (1998), "A shallow water equation as a geodesic flow on the Bott–Virasoro group", J. Geom. Phys. 24 (3): 203–208, Bibcode 1998JGP....24..203M, doi:10.1016/S0393-0440(97)00010-7
- Abenda, Simonetta; Grava, Tamara (2005), "Modulation of Camassa–Holm equation and reciprocal transformations", Ann. Inst. Fourier 55 (6): 1803–1834, arXiv:math-ph/0506042, Bibcode 2005math.ph...6042A, http://aif.cedram.org/item?id=AIF_2005__55_6_1803_0
- Alber, Mark S.; Camassa, Roberto; Holm, Darryl D.; Marsden, Jerrold E. (1994), "The geometry of peaked solitons and billiard solutions of a class of integrable PDEs", Lett. Math. Phys. 32 (2): 137–151, Bibcode 1994LMaPh..32..137A, doi:10.1007/BF00739423
- Alber, Mark S.; Camassa, Roberto; Holm, Darryl D.; Fedorov, Yuri N.; Marsden, Jerrold E. (2001), "The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type", Comm. Math. Phys. 221 (1): 197–227, arXiv:nlin/0105025, Bibcode 2001CMaPh.221..197A, doi:10.1007/PL00005573
- Artebrant, Robert; Schroll, Hans Joachim (2006), "Numerical simulation of Camassa–Holm peakons by adaptive upwinding", Applied Numerical Mathematics 56 (5): 695–711, doi:10.1016/j.apnum.2005.06.002
- Beals, Richard; Sattinger, David H.; Szmigielski, Jacek (2005), "Periodic peakons and Calogero–Françoise flows", J. Inst. Math. Jussieu 4 (1): 1–27, doi:10.1017/S1474748005000010
- Boutet de Monvel, Anne; Shepelsky, Dmitry (2005), "The Camassa–Holm equation on the half-line", C. R. Math. Acad. Sci. Paris 341 (10): 611–616, doi:10.1016/j.crma.2005.09.035
- Boutet de Monvel, Anne; Shepelsky, Dmitry (2006), "Riemann–Hilbert approach for the Camassa–Holm equation on the line", C. R. Math. Acad. Sci. Paris 343 (10): 627–632, doi:10.1016/j.crma.2006.10.014
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- Camassa, Roberto; Huang, Jingfang; Lee, Long (2005), "On a completely integrable numerical scheme for a nonlinear shallow-water wave equation", J. Nonlinear Math. Phys. 12 (suppl. 1): 146–162, Bibcode 2005JNMP...12S.146C, doi:10.2991/jnmp.2005.12.s1.13
- Camassa, Roberto; Huang, Jingfang; Lee, Long (2006), "Integral and integrable algorithms for a nonlinear shallow-water wave equation", J. Comput. Phys. 216 (2): 547–572, Bibcode 2006JCoPh.216..547C, doi:10.1016/j.jcp.2005.12.013
- Casati, Paolo; Lorenzoni, Paolo; Ortenzi, Giovanni; Pedroni, Marco (2005), "On the local and nonlocal Camassa–Holm hierarchies", J. Math. Phys. 46 (4): 042704, 8 pp., Bibcode 2005JMP....46d2704C, doi:10.1063/1.1888568
- Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl (2006), "A singular limit problem for conservation laws related to the Camassa–Holm shallow water equation", Comm. Partial Differential Equations 31 (7–9): 1253–1272, doi:10.1080/03605300600781600, http://www.math.ntnu.no/conservation/2005/017.html
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