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February  2021, 41(2): 967-985. 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  February 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (2) : 967-985. 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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