Dr. Antonio Degasperis

Revision as of 21:16, 4 January 2009

Degasperis-Procesi equation is a real nonlinear partial differential equation which models propagation of nonlinear dispersive waves and is solvable by the methods of soliton theory.

The DP equation

This is the partial differential equation (PDE)

<math DP>

u_t + c u_x + d u_{xxx}-a^2u_{xxt}-a^2f(uu_{xxx}+3u_xu_{xx})+4fu u_x=0 \,\,\,,\,\,\,u=u(x,t)\,\,, </math> for a function \(u(x,t)\) of the two variables \(x\) and \(t\). In this notation a subscripted variable indicates partial differentiation\[u_x \equiv \partial u /\partial x\,\,,\,\,u_{xx}\equiv \partial^2 u /\partial x^2\,\], etc.. This PDE remains the DP equation for any value of the real coefficients \(a\), \(c\), \(d\) and \(f\) with one exception\[a\] and \(f\) cannot be vanishing. On the other hand the factor 4 in front of the term \(u u_x\) and the ratio 3:1 of the coefficients of the terms \(u_x u_{xx}\) and \(u u_{xxx}\) cannot be changed as they are crucial to the special (and good) mathematical properties of the DP equation (<ref>DP</ref>). This PDE is not only of mathematical interest but it has also proved to be an approximate model of shallow water wave propagation in the small amplitude and long wavelength regime. Indeed, in this approximation, water is assumed to propagate in one direction over a flat bottom with no viscosity, no shear stress and no compressibility under the influence of gravity and surface tension. In this context, the dependent variable \(u\)is the horizontal velocity field while the independent variables \(x\) and \(t\) are the space coordinate and, respectively, the time. Also the coefficients have physical meaning\[c\] is the linear wave velocity, the coefficients \(a\) and \(d\) are related to linear dispersion and \(f\) originates from the Euler equation of motion. On the mathematical side the DP equation is very special because it belongs to the class of integrable equations, or solitonic equations (say PDEs with infinitely many conservation laws), for the investigation of which an analytical tool is available which generalizes to nonlinear PDEs the standard Fourier analysis of linear equations. In order to discover those PDEs which possess the special property of being integrable, a number of testing algorithms have been devised which provide necessary conditions for integrability. One particular test, based on perturbation theory, has been used to single all those PDEs out of the following family

<math bEQ>

u_t + c u_x + d u_{xxx}-a^2u_{xxt}-a^2f(uu_{xxx}+bu_xu_{xx})+gu u_x=0 \,\,\,,\,\,\,u=u(x,t)\,\,, </math> which satisfy the integrability conditions. Only three PDEs pass this test: the Korteweg de Vries (KdV) equation for \(a=0\), the Camassa-Holm (CH) equation for \(b=2\) and \(g=3f\) and the DP equation for \(b=3\) and \(g=4f\). This is how the DP equation has been first found while the other two equations, i.e. the KdV and CH equations, were already known to be integrable by different arguments. The DP equation (<ref>DP</ref>) can be rewritten in a cleaner form by the transformation \(u(x,t)\rightarrow u'(x,t)=\alpha u(\beta x+\gamma t,t) + \delta\) where the four parameters \(\alpha, \beta, \gamma,\delta\) can be so chosen that the coefficients of the DP equation (<ref>DP</ref>) for the new function \(u'(x,t)\) go into the values \(a=1\), \(c=0\), \(d=0\) and \(f=1\). With this choice of the coefficients the DP equation (<ref>DP</ref>) takes the neat form of the following system of two coupled differential equations

<math mDP>

m_t + u m_x + 3 u_{x}m=0 \,\,\,,\,\,\, u - u_{xx}=m\,\,. </math> This form of (<ref>DP</ref>) is more convenient to display its property of being an integrable equation. In fact, the distinctive feature of an integrable PDE is that of being the condition that two linear differential equations, which depend on an arbitrary complex parameter \(\lambda\), are compatible with each other. For the DP equation this pair of differential equations (generally referred to as Lax pair) reads

<math lax>

\begin{array}{lll} \psi_{xxx}&=&\psi_x+\lambda m(x,t) \psi\,\,,\\ \psi_t&=&\frac{1}{\lambda} \psi_{xx}-u(x,t) \psi_x + u_x(x,t) \psi\,\,, \end{array} </math> where the function \(\psi(x,t,\lambda)\) is the common solution of these two equations and \(\lambda\) is the auxiliary complex parameter (spectral variable). The two differential equations (<ref>lax</ref>) are compatible with each other for any value of \(\lambda\) if their coefficients \(m(x,t)\) and \(u(x,t)\) satisfy (<ref>mDP</ref>). Here the very existence and arbitrariness of the spectral parameter \(\lambda\) is of paramount importance in the method of investigation of the DP equation. These two differential equations (<ref>lax</ref>) have several consequences. An important one is that the DP equation has infinitely many conservation laws. Two separate sequences of them come out of the generating conservation law

<math gencons>

\rho_t(x,t,\lambda) =j_x(x,t,\lambda) \,\,, </math> where the density \(\rho(x,t,\lambda)\) and the current \(j(x,t,\lambda)\) have the \(\lambda\) dependent expression

<math roej>

\rho=(\textrm{log}\psi)_x \,\,\,,\,\,\, j=u_x-u\rho-\lambda^{-1} (\rho_x+\rho^2)\,\,. </math> One sequence of conservation laws is generated by expanding (<ref>gencons</ref>) in positive integer powers of the spectral variable \(\lambda\), while the other sequence obtains through the expansion of the same equation (<ref>gencons</ref>) in negative integer powers of \(\lambda^{1/3}\). The corresponding first few constants of the motion of the DP equation in the class of those solutions which vanish fast enough as \(x\rightarrow \pm \infty\) read

<math const>

\begin{array}{ll} H_{-1}=-\frac16 \int^{+\infty}_{-\infty} u^3 dx\,, & H_0=-\frac92 \int^{+\infty}_{-\infty} m dx\,, \,\,\, H_1=\frac12 \int^{+\infty}_{-\infty} mv dx\,, \\ H_5= \int^{+\infty}_{-\infty} m^{1/3} dx\,, & H_7=-\frac12 \int^{+\infty}_{-\infty} (m^{-7/3}m_x^2+9m^{-1/3}) dx\,, \end{array} </math> where the function \(v=v(x,t)\) is defined by the differential relation \(4v-v_{xx}=u\). The DP equation is also an infinite dimensional Hamiltonian system. In fact it can be written in Hamiltonian form in two different and independent ways:

<math hamilton>

m_t= B_0 \frac{\delta H_{-1}}{\delta m} \,\,,\,\, m_t= B_1 \frac{\delta H_{0}}{\delta m} \,, 

</math> where \(B_0 \) and \(B_1 \) are the skew-symmetric operators

<math sympl>

 B_0 =\partial_x (1-\partial_x^2)(4-\partial_x^2) \,\,,\,\, B_1=m^{2/3}\partial_x m^{1/3}(\partial_x-\partial_x^3)^{-1}m^{1/3}\partial_xm^{2/3}  \,. 

</math> The DP equation can also be written as the Euler derivative of a Lagrangian density \(\mathcal{L}\) or, equivalently, as the variational equation

<math var>

\delta \int \int \mathcal{L}dx dt=0 \,\,. </math>. To this purpose the DP equation (<ref>mDP</ref>) is more conveniently rewritten as the system

<math etaDP>

\eta_t + u \eta_x =0 \,\,\,,\,\,\, u - u_{xx}=\eta_x^3\,\,, </math> by the transformation \(m(x,t)=\eta_x^3(x,t)\), and, therefore, also as the single PDE

<math 1etaDP>

\frac{\eta_t}{\eta_x} - (\frac{\eta_t}{\eta_x})_{xx}+\eta_x^3=0\,\,,

</math> which is the variational condition (<ref>var</ref>) for the Lagrangian density function <math lagrange>

\mathcal{L}=\frac{\eta_t}{2\eta_x}[1+(\textrm{log}\eta_x )_{xx}]- \frac12 \eta_x^3\,\,.

</math>

Solutions

As for any wave propagation model, the main task is that of investigating the solution \(u(x,t)\) for \(t>0\) which satisfies the initial condition \(u(x,0)=u_0(x)\) at \(t=0\) where \(u_0(x)\) is a given profile. The way to approach this problem depends on the class of initial data \(u_0(x)\). Solutions \(u(x,t)\) exist such they remain smooth (i.e. everywhere continuous with continuous derivatives) in the variable \(x\) at any time \(t>0\) if the initial value \(u_0(x)\) is smooth. However, even if the initial profile \(u_0(x)\) is smooth but satisfies appropriate conditions, the corresponding solution \(u(x,t)\) develops a singularity at a finite critical time \(t=T_c>0\), namely its first derivative \(u_x(x,t)\) becomes infinitely large at a point, as for a shock wave, and a wave breaking process takes place. Some of these blowing up solutions can be extended after the critical time namely for \(t>T_c\). With the exception of the special class of the so--called peakon solutions (see below), the initial value problem for the DP equation has not been approached yet by making use of the pair of linear equations (<ref>lax</ref>) in both cases of smooth solutions and the so--called weak solutions. These are solutions which are distributions rather than ordinary functions. In this context, existence and uniqueness of (entropy) weak solutions have been proved for a special class of non smooth initial values \(u_0(x)\). The richness of the scenario of solutions of the DP equation is already displayed by a variety of special solutions which have been analytically constructed. Such is the N--soliton solution which describes the nonlinear superposition and collision of N localized special waves (solitons). Its expression is implicitly known; for instance the one soliton solution of the DP equation has the parametric representation \(u(x,t)=A(\xi)\) and \(x=B(\xi,t)\) where \(-\infty<\xi<+\infty\) is the real parameter and

<math soliton>

\begin{array}{lll} A(\xi)&=&b^3+3b^3\frac{(a^2-1)(4a^2-1)}{2a^2-1+a\cosh(\xi)}\,\,,\\ B(\xi,t)&=&\frac{\xi}{p} +4a^2 b^3 t +\log\left[\frac{\alpha+1+e^{\xi}(\alpha-1)}{\alpha-1+e^{\xi}(\alpha+1)}\right] +x_0\,\,. \end{array} </math> The free parameters are \(p, b\) and \(x_0\) while \(a=[(1-p^2/4)/(1-p^2)]^{1/2}\) and \(\alpha=\{[(2a-1)(a+1)]/[(2a+1)(a-1)]\}^{1/2}\). This soliton is a localized wave on the flat background \(u(\pm \infty ,t)=b^3\). For particular values of the parameters this solution \(u(x,t)\) is multivalued and it is generally referred to as loop--soliton solution. Important examples of weak solutions of the DP equation (<ref>mDP</ref>) are the N--shockpeakon solutions which describe the collision of N discontinuous profiles. Their general expression reads

<math shock>

u(x,t)=\sum_{j=i}^N[p_j(t)-sign(x-q_j(t))s_j(t)]e^{-|x-q_j(t)|}\,\,,\,\,m(x,t)=2\sum_{j=i}^N[p_j(t)\delta(x-q_j(t))+s_j(t)\delta^{(1)}(x-q_j(t))]\,\,, </math> where \(\delta(x)\) is the Dirac distribution, \(\delta^{(1)}(x)\) is its first derivative, and the dynamical variables \(q_j(t), p_j(t), s_j(t)\) solve the following system of ordinary differential equations

<math speakon>

\begin{array}{lll} \frac{dq_j}{dt}&=&\sum_{n=1}^N[p_n-sign(q_j-q_n)s_n]e^{-|q_j-q_n|}\,\,,\\ \frac{dp_j}{dt}&=&2\sum_{n=1}^N[p_n-sign(q_j-q_n)s_n][s_j+sign(q_j-q_n)p_j]e^{-|q_j-q_n|}\,\,\\ \frac{ds_j}{dt}&=&-s_j\sum_{n=1}^N[s_n-sign(q_j-q_n)p_n]e^{-|q_j-q_n|}\,\,. \end{array} </math> In particular, the one shockpeakon solution is the expression (<ref>shock</ref>) for N=1 and \(q_1(t)=p_0t+q_0, \,p_1(t)=p_0, \,s_1(t)=s_0/(1+s_0t)\) where \(q_0, \,p_0 \) are arbitrary and \(s_0>0 \) is positive. The subclass of these solutions which are characterized by the condition \(s_j(t)=0\) are known as N--peakon solutions because \(u(x,t)\) shows a peak at its maxima in \(x\) where \(u(x,t)\) is continuous but its first derivative \(u_x(x,t)\) is discontinuous. The N--peakon solution has been constructed by making use of the pair of linear equations (<ref>lax</ref>), while the general solution the N--shockpeakon solution of the system (<ref>speakon</ref>) is still not known for N>1. Other known solutions of the DP equation are the traveling wave solutions of the form \(u(x,t)=v(x-ct)\). They have been completely classified and constructed in the class of both smooth and weak solutions, and as well in the periodic and localized cases. Some of these solutions are obtained by gluing together smooth solutions and, according to their resulting shape, they have been given names such as cuspons and stumpons.

References

• [DP99] Degasperis A., Procesi M. Asymptotic Integrability, Symmetry and Perturbation Theory (Rome, 1998) (A. Degasperis and G. Gaeta, eds.), World Scientific Publishing, New Jersey, 1999, pp. 23-37

• [DHH02] Degasperis A., Holm D. D., Hone A. N. W. A new integrable equation with peakons solutions, Theoret. and Math. Phys. 133 1463-1474 (2002)

• [MN02] Mikhailov A. V., Novikov V. S. Perturbative symmetry approach, J. Phys. A: Math. Gen. 35 4775-4790 (2002)

• [DGH03] Dullin H. R., Gottwald G. A., Holm D. D. Camassa-Holm, Korteweg-de Vries-5and other asymptotically equivalent equations for shallow water wave, Fluid Dynamics Research 33 73-95 (2003)

• [J03] Johnson R. S. The classical problem of water waves: a reservoir of integrable and nearly-integrable equations, J. Nonlin. Math. Phys. 10(Supplement 1): 72–92 (2003)

• [LS03] Lundmark H., Szmigielski J. Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems 19 1241-1245 (2003)
• [Y03-1] Yin Zhaoyang Global existence for a new periodic integrable equation, J. Math. Anal. Appl. 283 129–139 (2003)

• [Y03-2] Yin Zhaoyang On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47 649–666 (2003)

• [DGH04] Dullin H. R., Gottwald G. A., Holm D. D. On asymptotically equivalent shallow water wave equations, Physica D 190 1–14 (2004)

• [VP04-1] Vakhnenko V.O., Parkes E.J. The connection of the Degasperis-Procesi equation with the Vakhnenko equation, Proceedings of Institute of Mathematics of NAS of Ukraine 50 493-497 (2004)

• [VP04-2] Vakhnenko V.O., Parkes E.J. Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos, Solitons & Fractals, 20 1059-1073(2004)

• [Y04-1] Yin Zhaoyang Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J. 53 1189–1209 (2004)

• [Y04-2] Yin Zhaoyang Global weak solutions for a new periodic integrable equation with peakon solution, J. Funct. Anal. 212 182–194 (2004)

• [H05] Henry D. Infinite propagation speed for the Degasperis–Procesi equation, J. Math. Anal. Appl. 311 755-759 (2005)

• [I05] Ivanov R. On the integrability of a class of nonlinear dispersive wave equations, J.Nonlin. Math. Phys. 12 462–468 (2005)

• [L05] Lenells J. Traveling wave solutions of the Degasperis-Procesi equation,J. Math. Anal. Appl. 306 72-82 (2005)

• [LS05] Lundmark H., Szmigielski J. Degasperis-Procesi peakons and the discrete cubic string, International Mathematics Research Papers 53-116 (2005)

• [KLM05] Kraenkel R.A., Leon J., Manna M.A. Theory of small aspect ratio waves in deep water, Physica D 211 377-399 (2005)

• [M05-1] Matsuno Y. Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit, Inverse Problems 21 1553-1570 (2005)

• [M05-2] Matsuno Y. The N-soliton solution of the Degasperis-Procesi equation,Inverse Problems 21 2085-2101 (2005)

• [Mu05] Mustafa O.G. A note on the Degasperis-Procesi equation, J. Nonlin. Math. Phys. 12 10–14 (2005)

• [CK06] Coclite G.M., Karlsen K.H. On the well-posedness of the Degasperis-Procesi equation, Journal of Functional Analysis 233 60-91 (2006)

• [ELY06] Escher J., Liu Yue, Yin Zhaoyang Global weak solutions and blow-up structure for the Degasperis–Procesi equation, J. Funct. Anal. 241 457–48 (2006)

• [LY06] Liu Yue. Yin Zhaoyang Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys. 267 801–820 (2006)

• [M06] Matsuno Y. Cusp and loop soliton solutions of short wave models for the Camassa- Holm and Degasperis-Procesi equations, Phys. Lett. A 359 451-457 (2006)

• [CL07] Constantin A., Lannes D. The hydrodinamical relenvance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal. Doi 10.1007/s00205-008-0128-2 (2007)<http://arxiv.org/abs/0709.0905>

• [CK07] Coclite G.M., Karlsen K.H. On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation, J. Differential Equations 234 142-160 (2007)

• [E07] Escher J. Wave breaking and shock waves for a periodic shallow water equation, Phil. Trans. R. Soc. A 365 2281–2289 (2007)

• [ELY07-1] Escher J., Liu Yue, Yin Zhaoyang Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation Indiana Univ. Math. J. 56 87–117(2007)

• [EY07] Escher J., Yin Zhaoyang On the initial boundary value problems for the Degasperis-Procesi equation, Phys. Lett. A 368 69–76 (2007)

• [G07] Partha Guha Euler-Poincaré formalism of (two component) Degasperis-Procesi and Holm-Staley type systems, J. Nonlin. Math. Phys. 14 398–429 (2007)

• [H07] Hoel H. A. A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation, Electron. J. Differential Equations 2007 No. 100 1–10 (2007)

• [I07] Ivanov R. Water waves and integrability, Phil. Trans. R. Soc. A 365 2267–2280 (2007)

Links