Cheng, Feifei; Li, Ji Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. (English) Zbl 1477.34081 Discrete Contin. Dyn. Syst. 41, No. 2, 967-985 (2021). By using the geometric singular perturbation theory, the authors prove the existence of solitary wave solutions for the Degasperis-Procesi equation with distributed delay. The main idea is to transform the Degasperis-Procesi equation to a slow-fast system and then prove this system possesses a homoclinic orbit which corresponds to a solitary wave solution of the original equation. Reviewer: Xiang-Sheng Wang (Lafayette) Cited in 5 Documents MSC: 34E15 Singular perturbations for ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 35C07 Traveling wave solutions Keywords:Degasperis-Procesi equation; solitary wave solutions; geometric singular perturbation theory; invariant manifold; homoclinic orbits PDFBibTeX XMLCite \textit{F. Cheng} and \textit{J. Li}, Discrete Contin. Dyn. Syst. 41, No. 2, 967--985 (2021; Zbl 1477.34081) Full Text: DOI References: [1] A. Constantin; J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025 [2] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50, 321-362 (2000) · Zbl 0944.00010 [3] A. Constantin; B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78, 787-804 (2003) · Zbl 1037.37032 [4] A. Constantin; D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010 [5] R. Camassa; D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521 [6] A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, (1999), 23-37. · Zbl 0963.35167 [7] A. Degasperis; D. D. Holm; A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133, 1463-1474 (2002) [8] Z. Du; J. Li; X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Functional Analysis, 275, 988-1007 (2018) · Zbl 1392.35223 [9] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31, 53-98 (1979) · Zbl 0476.34034 [10] D. D. Holm; M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2, 323-380 (2003) · Zbl 1088.76531 [11] G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60, 347-386 (2010) · Zbl 1311.34133 [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. · Zbl 0840.58040 [13] D. J. Korteweg; 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, 422-443 (1895) · JFM 26.0881.02 [14] Y. Li; P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119 [15] Y. Liu; Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267, 801-820 (2006) · Zbl 1131.35074 [16] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24, 203-208 (1998) · Zbl 0901.58022 [17] C. Robinson, Sustained resonance for a nonlinear system with slowly varying coefficients, SIAM J. Math. Anal., 14, 847-860 (1983) · Zbl 0523.34035 [18] P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations, 92, 252-281 (1991) · Zbl 0734.34038 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.