Degasperis-Procesi equation

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Antonio Degasperis and Michela Procesi (2009), Scholarpedia, 4(2):7318. doi:10.4249/scholarpedia.7318 revision #91190 [link to/cite this article]
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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.

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The DP equation

This is the partial differential equation (PDE) \[\tag{1} u_t + c u_x + d u_{xxx}-a^2u_{xxt}-a^2f(uu_{xxx}+3u_xu_{xx})+4fu u_x=0 \,\,\,,\,\,\,u=u(x,t)\,\,, \]

for a real function \(u(x,t)\) of the two variables \(x\) and \(t\ .\) In this notation a subscripted variable indicates partial differentiation\[u_x \equiv \partial u /\partial x\,\,,\,\,u_{xt}\equiv \partial^2 u /\partial x\partial t\,\ ,\] etc.. The DP equation has real coefficients \(a\ ,\) \(c\ ,\) \(d\) and \(f\) with \(af\neq 0\ ,\) and the form of the equation is covariant with respect to a combination of scaling, shiftingand Galilei transformations (see below). 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 (1). 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 (Johnson 2003, Dullin, Gottwald and Holm 2004, Constantin and Lannes 2007, Ivanov 2007). Indeed, in this approximation, waves are 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 (that is, 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 (Degasperis 2008), based on multiscale perturbation theory, has been used to single all those PDEs, which satisfy the integrability conditions, out of the following family \[\tag{2} u_t + c u_x + d u_{xxx}-a^2u_{xxt}-a^2f(uu_{xxx}+bu_xu_{xx})+gu u_x=0 \,\,\,,\,\,\,u=u(x,t)\,\,. \]

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 (Degasperis and Procesi 1999) while the other two equations, i.e. the KdV and CH equations, were already known to be integrable by different arguments. The DP equation (1) is covariant under the group of transformations \(u(x,t)\rightarrow u'(x,t)=\alpha u(\beta x+\gamma t,t) + \delta\ ,\) and by a suitable choice of the parameters \(\alpha, \beta, \gamma,\delta\) the coefficients can be fixed as \(a=1\ ,\) \(c=0\ ,\) \(d=0\) and \(f=1\ .\) With this choice of the coefficients (after dropping the prime) the DP equation (1) takes the neat form of the following system of two coupled differential equations \[\tag{3} m_t + u m_x + 3 u_{x}m=0 \,\,\,,\,\,\, u - u_{xx}=m\,\,. \]

This form of (1) 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 (Degasperis, Hone and Holm 2002) \[\tag{4} \begin{array}{lll} \psi_{xxx}&=&\psi_x+\lambda m(x,t) \psi\,\,,\\ \psi_t&=&\frac{1}{\lambda} \psi_{xx}-u(x,t) \psi_x + u_x(x,t) \psi\,\,, \end{array} \]

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 (4) are compatible with each other for any value of \(\lambda\) if their coefficients \(m(x,t)\) and \(u(x,t)\) satisfy (3). 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 (4) 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 \[\tag{5} \rho_t(x,t,\lambda) =j_x(x,t,\lambda) \,\,, \]

where the density \(\rho(x,t,\lambda)\) and the current \(j(x,t,\lambda)\) have the \(\lambda\) dependent expression \[\tag{6} \rho=(\textrm{log}\psi)_x \,\,\,,\,\,\, j=u_x-u\rho-\lambda^{-1} (\rho_x+\rho^2)\,\,. \]

One sequence of conservation laws is generated by expanding (5) in positive integer powers of the spectral variable \(\lambda\ ,\) while the other sequence is obtained through the expansion of the same equation (5) 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 \[\tag{7} \begin{array}{ll} H_{-1}=-\frac16 \int^{+\infty}_{-\infty} u^3 dx\,, & H_0=-\frac92 \int^{+\infty}_{-\infty} m dx\,, \,\,\, H_1=\frac12 \int^{+\infty}_{-\infty} mv dx\,, \\ H_5= \int^{+\infty}_{-\infty} m^{1/3} dx\,, & H_7=-\frac12 \int^{+\infty}_{-\infty} (m^{-7/3}m_x^2+9m^{-1/3}) dx\,, \end{array} \]

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: \[\tag{8} m_t= B_0 \frac{\delta H_{-1}}{\delta m} \,\,,\,\, m_t= B_1 \frac{\delta H_{0}}{\delta m} \,, \]

where \(B_0 \) and \(B_1 \) are the skew-symmetric operators \[\tag{9} B_0 =\partial_x (1-\partial_x^2)(4-\partial_x^2) \,\,,\,\, B_1=m^{2/3}\partial_x m^{1/3}(\partial_x-\partial_x^3)^{-1}m^{1/3}\partial_xm^{2/3} \,. \]

The operators \(B_0\) and \(B_1\) form a compatible bi-Hamiltonian pair (Hone and Wang 2003). The DP equation can also be written as the Euler derivative of a Lagrangian density \(\mathcal{L}\) or, equivalently, as the variational equation \[\tag{10} \delta \int \int \mathcal{L}dx dt=0 \,\,. \ .\]

To this purpose the DP equation (3) is more conveniently rewritten as the system \[\tag{11} \eta_t + u \eta_x =0 \,\,\,,\,\,\, u - u_{xx}=\eta_x^3\,\,, \]

by the transformation \(m(x,t)=\eta_x^3(x,t)\ ,\) and, therefore, also as the single PDE \[\tag{12} \frac{\eta_t}{\eta_x} - (\frac{\eta_t}{\eta_x})_{xx}+\eta_x^3=0\,\,, \]

which is the variational condition (10) for the Lagrangian density function \(\tag{13} \mathcal{L}=\frac{\eta_t}{2\eta_x}[1+(\textrm{log}\eta_x )_{xx}]- \frac12 \eta_x^3\,\,. \)


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 that 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, if it satisfies additional technical conditions, then 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 (Liu and Yin 2006). 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 yet been approached by making use of the pair of linear equations (4) in both cases of smooth and of non smooth, so-called weak, solutions. These are solutions which are distributions rather than ordinary functions. In this context, existence and uniqueness of weak solutions have been proved for a special class of non smooth initial values \(u_0(x)\) (Escher, Liu and Yin 2006, Coclite and Karlsen 2007). The initial value problem associated with the DP equation has been mainly investigated for \(x\) in the whole real axis with appropriate asymptotic conditions. Further analysis has been also done of the initial boundary value problem for \(x\) in a semi--axis, as well as for \(x\) in an interval of the real axis (Escher and Yin 2007). 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 (Matsuno 2005) which describes the nonlinear superposition and collision of N localized special waves, i.e. solitons (for an introduction to the literature see Degasperis 1998 ). Its expression is implicitly known; for instance the one soliton solution of the DP equation (3) has the parametric representation \(u(x,t)=A(\xi)\) and \(x=B(\xi,t)\) where \(-\infty<\xi<+\infty\) is the real parameter and \[\tag{14} \begin{array}{lll} A(\xi)&=&b^3+3b^3\frac{(a^2-1)(4a^2-1)}{2a^2-1+a\cosh(\xi)}\,\,,\\ B(\xi,t)&=&\frac{\xi}{p} +4a^2 b^3 t +\log\left[\frac{\alpha+1+e^{\xi}(\alpha-1)}{\alpha-1+e^{\xi}(\alpha+1)}\right] +x_0\,\,. \end{array} \]

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 (3) are the N-shockpeakon solutions (Lundmark 2007) which describe the collision of N discontinuous profiles. Their general expression reads \[\tag{15} \begin{array}{lll} u(x,t)&=&\sum_{j=i}^N[p_j(t)-sign(x-q_j(t))s_j(t)]e^{-|x-q_j(t)|}\,\,,\\m(x,t)&=&2\sum_{j=i}^N[p_j(t)\delta(x-q_j(t))+s_j(t)\delta^{(1)}(x-q_j(t))]\,\,, \end{array} \]

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 \[\tag{16} \begin{array}{lll} dq_j/dt&=&\sum_{n=1}^N[p_n-sign(q_j-q_n)s_n]e^{-|q_j-q_n|}\,\,,\\ dp_j/dt&=&2\sum_{n=1}^N[p_n-sign(q_j-q_n)s_n][s_j+sign(q_j-q_n)p_j]e^{-|q_j-q_n|}\,\,\\ ds_j/dt&=&-s_j\sum_{n=1}^N[s_n-sign(q_j-q_n)p_n]e^{-|q_j-q_n|}\,\,. \end{array} \]

In particular, the one--shockpeakon solution is the expression (15) 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 and finite. The N--peakon dynamical system, whose equations of motion are (16) with \(s_j(t)=0\ ,\) is Hamiltonian (Degasperis, Hone and Holm 2003). Its solutions have been constructed by making use of the pair of linear equations (4) (Lundmark and Szmigielski 2005), and its stability has been established ( Lin and Liu 2008). However the solution of the N--shockpeakon dynamical system (16) with \(s_j(t)\neq 0\) 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 classified (Lenells 2005, Lenells 2007) and some of them have been constructed explicitly (Qiao 2008) in both the classes of 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

  • Coclite G.M., Karlsen K.H. On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation, J. Differential Equations 234 (2007) 142-160
  • Constantin A., Lannes D. The hydrodinamical relenvance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal. (2008) Doi 10.1007/s00205-008-0128-2
  • Degasperis A. Solitons, Am. J. Phys. 66 (1998) 486-497
  • Degasperis A., Procesi M. Asymptotic Integrability, in Symmetry and Perturbation Theory (A. Degasperis and G. Gaeta, eds.), World Scientific Publishing, 1999, 23-37
  • Degasperis A., Holm D. D., Hone A. N. W. A new integrable equation with peakon solutions, Theoret. and Math. Phys. 133 (2002) 1463-1474
  • Degasperis A., Holm D. D., Hone A. N. W. Integrable and non--integrable equations with peakons, in Nonlinear Physics: Theory and Experiment. II ( M.J. Ablowitz, M. Boiti, F. Pempinelli and B. Prinari, eds.), World Scientific Publishing, 2007, 37-43
  • Degasperis A. Multiscale Expansion and Integrability of Dispersive Wave Equations, in Integrability (A. Mikhailov, ed.) Lecture Notes in Physics 767 , Springer, 2008, 215-244
  • Dullin H. R., Gottwald G. A., Holm D. D. On asymptotically equivalent shallow water wave equations, Physica D 190 (2004) 1–14
  • Escher J., Liu Yue, Yin Zhaoyang Global weak solutions and blow-up structure for the Degasperis–Procesi equation, J. Funct. Anal. 241 (2006) 457–48
  • Escher J., Yin Zhaoyang On the initial boundary value problems for the Degasperis-Procesi equation, Phys. Lett. A 368 (2007) 69–76
  • Hone A.N.W. and Wang Jing Ping Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems 19 (2003) 129-145
  • Ivanov R. Water waves and integrability, Phil. Trans. R. Soc. A 365 (2007) 2267–2280
  • Johnson R. S. The classical problem of water waves: a reservoir of integrable and nearly-integrable equations, J. Nonlin. Math. Phys. 10 (Supplement 1) (2003) 72–92
  • Lenells J. Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl. 306 (2005) 72-82
  • Lenells J. Classification of all travelling-wave solutions for some nonlinear dispersive equations, Phil. Trans. R. Soc. A 365 (2007) 2291–2298
  • Lin Zhiwu, Liu Yue Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62 (2008) 125-146
  • Liu Yue. Yin Zhaoyang Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys. 267 (2006) 801–820
  • Lundmark H., Szmigielski J. Degasperis-Procesi peakons and the discrete cubic string, International Mathematics Research Papers (2005) No.2 53-116
  • Lundmark H. Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci. 17 (2007) 169-198
  • Matsuno Y. The N-soliton solution of the Degasperis-Procesi equation, Inverse Problems 21 (2005) 2085-2101
  • Qiao Zhijun M-Shape peakons, dehisced solitons, cuspons and new 1-peak solitons for the Degasperis-Procesi equation, Chaos Solitons Fractals 37 (2008) 501-507


Internal references

  • Mark Ablowitz and Barbara Prinari (2008) Nonlinear Schrodinger systems: continuous and discrete. Scholarpedia, 3(8): 5561

See also

Shallow water waves, Nonlinear waves, Multiscale expansion method, Soliton, Integrable system, Inverse scattering transform, Korteweg-de Vries equation, Camassa-Holm equation

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