In this paper, we consider the solution map of the Cauchy problem to the Fokas–Olver–Rosenau–Qiao equation on the real line and prove that the solution map of this problem is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous results in Himonas and Mantzavinos (Nonlinear Anal 95:499–529, 2014) and Li et al. (J Math Fluid Mech 22:50, 2020).
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Bahouri, H., Chemin, J., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (2011)
Chang, X., Szmigielski, J.: Lax integrability and the peakon problem for the modified Camassa–Holm equation. Comm. Math. Phys. 358, 295–341 (2018)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Constantin, A.: The Hamiltonian structure of the Camassa–Holm equation. Expo. Math. 15, 53–85 (1997)
Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)
Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)
Constantin, A., Molinet, L.: Orbital stability of solitary waves for a shallow water equation. Phys. D 157, 75–89 (2001)
Constantin, A., Strauss, W.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000)
Fokas, A.: On a class of physically important integrable equations. Phys. D 87, 145–150 (1995)
Fu, Y., Gui, G., Liu, Y., Qu, C.: On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity. J. Differ. Equ. 255, 1905–1938 (2013)
Fuchssteiner, B.: Some tricks from the symmetry toolbox for nonlinear equations: generalisations of the Camassa–Holm equation. Phys. D 95, 229–243 (1996)
Gui, G., Liu, Y., Olver, P., Qu, C.: Wave-breaking and peakons for a modified Camassa–Holm equation. Commun. Math. Phys. 319, 731–759 (2013)
Himonas, A., Misiołek, G.: High-frequency smooth solutions and well-posedness of the Camassa–Holm equation. Int. Math. Res. Not. 51, 3135–3151 (2005)
Himonas, A., Kenig, C.: Non-uniform dependence on initial data for the CH equation on the line. Diff. Integr. Equ. 22, 201–224 (2009)
Himonas, A., Kenig, C., Misiołek, G.: Non-uniform dependence for the periodic CH equation. Commun. Partial Differ. Equ. 35, 1145–1162 (2010)
Himonas, A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)
Himonas, A., Mantzavinos, D.: The Cauchy problem for the Fokas–Olver–rosenau–Qiao equation. Nonlinear Anal. 95, 499–529 (2014)
Himonas, A., Matzavinos, D.: Hölder continuity for the Fokas–Olver–Rosenau–Qiao equation. J. Nonlinear Sci. 24, 1105–1124 (2014)
Himonas, A., Holliman, C.: Non-uniqueness for the Fokas–Olver–Rosenau–Qiao equation. J. Math. Anal. Appl. 470, 647–658 (2019)
Holmes, J., Tiglay, F., Thompson, R.: Continuity of the data-to-solution map for the FORQ equation in Besov spaces. Differ. Integral Equ. 34(5/6), 295–314 (2021)
Kenig, C., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. 106, 617–633 (2001)
Li, J., Yin, Z.: Well-posedness and analytic solutions of the two-component Euler–Poincaré system. Monatsh. Math. 183, 509–537 (2017)
Li, J., Yu, Y., Zhu, W.: Non-uniform dependence on initial data for the Camassa–Holm equation in Besov spaces. J. Differ. Equ. 269, 8686–8700 (2020)
Li, J., Li, M., Zhu, W.: Non-uniform dependence for Novikov equation in Besov spaces. J. Math. Fluid Mech. 22(4), 50 (2020)
Liu, X., Liu, Y., Qu, C.: Orbital stability of the train of peakons for an integrable modified Camassa–Holm equation. Adv. Math. 255, 1–37 (2014)
Olver, P.J., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)
Qiao, Z.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006)
Qiao, Z.: New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W shape peak solitons. J. Math. Phys. 48, 082701 (2007)
Qu, C., Liu, X., Liu, Y.: Stability of peakons for an integrable modified Camassa–Holm equation with cubic nonlinearity. Comm. Math. Phys. 322, 967–997 (2013)
The authors want to thank the referees for their careful reading and helpful suggestions, which greatly improved the presentation of this paper. The authors are very grateful to Dr. Jinlu Li for some useful suggestions. Y. Yu is supported by the Natural Science Foundation of Anhui Province (No. 1908085QA05). This work is partially supported by the National Natural Science Foundation of China (Grant No. 11801090).
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Communicated by Adrian Constantin.
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Wu, X., Yu, Y. Non-uniform continuity of the Fokas–Olver–Rosenau–Qiao equation in Besov spaces. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01637-2
- Fokas–Olver–Rosenau–Qiao equation
- Non-uniform continuous dependence
- Besov spaces
Mathematics Subject Classification