doi: 10.3934/dcds.2020305

Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Ji Li, liji@hust.edu.cn

Received  April 2020 Published  August 2020

Fund Project: The corresponding author is supported by NSFC grant 11771161

In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.

Citation: Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020305
References:
[1]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[2]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[3]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[4]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, (1999), 23–37.  Google Scholar

[7]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422.  Google Scholar

[8]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Functional Analysis, 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[9]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[10]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.  Google Scholar

[11]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[12]

C. K. R. T. Jones, Geometrical singular perturbation theory, Lecture Notes in Mathematics: Dynamical systems (eds. R. Johnson), Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[13]

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., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[14]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[15]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[16]

G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[17]

C. Robinson, Sustained resonance for a nonlinear system with slowly varying coefficients, SIAM J. Math. Anal., 14 (1983), 847-860.  doi: 10.1137/0514066.  Google Scholar

[18]

P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations, 92 (1991), 252-281.  doi: 10.1016/0022-0396(91)90049-F.  Google Scholar

show all references

References:
[1]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[2]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[3]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[4]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, (1999), 23–37.  Google Scholar

[7]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422.  Google Scholar

[8]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Functional Analysis, 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[9]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[10]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.  Google Scholar

[11]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[12]

C. K. R. T. Jones, Geometrical singular perturbation theory, Lecture Notes in Mathematics: Dynamical systems (eds. R. Johnson), Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[13]

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., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[14]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[15]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[16]

G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[17]

C. Robinson, Sustained resonance for a nonlinear system with slowly varying coefficients, SIAM J. Math. Anal., 14 (1983), 847-860.  doi: 10.1137/0514066.  Google Scholar

[18]

P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations, 92 (1991), 252-281.  doi: 10.1016/0022-0396(91)90049-F.  Google Scholar

Figure 1.  The homoclinic orbit within $ \phi<c-\frac{2}{3}k $
Figure 2.  local delay, k = 1
Figure 3.  noncocal delay, k = 1
[1]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[3]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[4]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[5]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[6]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[7]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[8]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[9]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[11]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[12]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[13]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[14]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[15]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[16]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[17]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[18]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[19]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[20]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (57)
  • HTML views (155)
  • Cited by (0)

Other articles
by authors

[Back to Top]