An efficient method to compute solitary wave solutions of fractional Korteweg–de Vries equations

References

• J. Alvarez and A. Durán, Petviashvili type methods for traveling wave computations: I. Analysis of convergence, J. Comput. Appl. Math. 266 (2014), pp. 39–51. doi: 10.1016/j.cam.2014.01.015
   Web of Science ®Google Scholar
• J. Alvarez and A. Durán, Petviashvili type methods for traveling wave computations: II. Acceleration with vector extrapolation methods, Math. Comput. Simul. 123 (2016), pp. 19–36. doi: 10.1016/j.matcom.2015.10.015
   Web of Science ®Google Scholar
• J. Alvarez and A. Durán, Numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems, J. Comput. Appl. Math. 316 (2017), pp. 29–39. doi: 10.1016/j.cam.2016.08.037
   Web of Science ®Google Scholar
• J. Angulo Pava, Nonlinear Dispersive Equations, Existence and Stability of Solitary and Periodic Traveling Wave Solutions, Mathematical Surveys and Monographs, Vol. 156, AMS, Providence, RI, 2009.
   Google Scholar
• J. Angulo Pava, Stability properties of solitary waves for fractional KdV and BBM equations, preprint (2017). Available at http://www.arxiv.org/abs/1701.06221.
   Google Scholar
• M.N. Arnesen, Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst. 36 (2016), pp. 3483–3510. doi: 10.3934/dcds.2016.36.3483
   Web of Science ®Google Scholar
• J.L. Bona, V.A. Dougalis, O.A. Karakashian, and W.R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Phil. Trans. R. Soc. London A 351 (1995), pp. 107–164. doi: 10.1098/rsta.1995.0027
   Web of Science ®Google Scholar
• J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, New York, 2000.
   Google Scholar
• C. Brezinski, Convergence acceleration during the 20th century, J. Comput. Appl. Math. 122 (2000), pp. 1–21. doi: 10.1016/S0377-0427(00)00360-5
   Web of Science ®Google Scholar
• C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, Heidelberg, Berlin, 1988.
   Google Scholar
• E. Dinvay, D. Moldabayev, D. Dutykh, and H. Kalisch, The Whitham equation with surface tension, Nonlinear Dyn. 88 (2017), pp. 1125–1138. doi: 10.1007/s11071-016-3299-7
   Web of Science ®Google Scholar
• V.A. Dougalis, A. Durán, M.A. López-Marcos, and D.E. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith family of Boussinesq systems, J. Nonlinear Sci. 17 (2007), pp. 569–607. doi: 10.1007/s00332-007-9004-8
   Web of Science ®Google Scholar
• M. Ehrnström, M.D. Groves, and E. Wahlen, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity 25 (2012), pp. 2903–2936. doi: 10.1088/0951-7715/25/10/2903
   Web of Science ®Google Scholar
• P. Felmer, A. Quaas, and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. A 142 (2012), pp. 1237–1262. doi: 10.1017/S0308210511000746
   Google Scholar
• R.L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in R, Acta Math. 210 (2013), pp. 261–318. doi: 10.1007/s11511-013-0095-9
   Web of Science ®Google Scholar
• J. de Frutos and J.M. Sanz-Serna, An easily implementable fourth-order method for the time integration of wave problems, J. Comput. Phys. 103 (1992), pp. 160–168. doi: 10.1016/0021-9991(92)90331-R
   Web of Science ®Google Scholar
• M. Grillakis, J. Shatah, and W.A. Strauss, Stability theory of solitary waves in the presence of symmetry: I, J. Funct. Anal. 74 (1987), pp. 160–197. doi: 10.1016/0022-1236(87)90044-9
   Web of Science ®Google Scholar
• E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, New York, Heidelberg, Berlin, 2004.
   Google Scholar
• S. Herr, A.D. Ionescu, C.E. Kenig, and H. Koch, A para-differential renormalization technique for nonlinear dispersive equations, Comm. Partial Diff. Equ. 35 (2010), pp. 1827–1875. doi: 10.1080/03605302.2010.487232
   Web of Science ®Google Scholar
• K. Jbilou and H. Sadok, Vector extrapolation methods. Applications and numerical comparison, J. Comput. Appl. Math. 122 (2000), pp. 149–165. doi: 10.1016/S0377-0427(00)00357-5
   Web of Science ®Google Scholar
• H. Kalisch, D. Moldabayev, and O. Verdier, A numerical study of nonlinear dispersive wave models with SpecTraWave, Electron. J. Diff. Equ. 62 (2017), pp. 1–23.
   Google Scholar
• T. Kapitula and A. Stefanov, A Hamiltonia-Krein (instability) index theory for solitary waves to KdV-like eigenvalue problems, Stud. Appl. Math. 132 (2014), pp. 183–211. doi: 10.1111/sapm.12031
   Web of Science ®Google Scholar
• C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burges' equation, Phys. D 295–296 (2015), pp. 46–65. doi: 10.1016/j.physd.2014.12.004
   Web of Science ®Google Scholar
• D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Survey and Monographs, Vol. 188, AMS, Providence, RI, 2013.
   Google Scholar
• D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation, Kinet. Related Models AIMS 6(4) (2013), pp. 989–1009. doi: 10.3934/krm.2013.6.989
   Web of Science ®Google Scholar
• F. Linares, D. Pilod, and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal. 46(2) (2014), pp. 1505–1537. doi: 10.1137/130912001
   Web of Science ®Google Scholar
• F. Linares, D. Pilod, and J.-C. Saut, Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Diff. Equ. 20 (2015), pp. 835–858.
   Web of Science ®Google Scholar
• Y. Martel, F. Merle, and V Raphaël, Blow up for the critical gKdV equation I: Dynamics near the soliton, Acta Math. 212 (2014), pp. 27–45. doi: 10.1007/s11511-014-0109-2
   Web of Science ®Google Scholar
• Y. Martel, F. Merle, and P Raphaël, Blow up for the critical gKdV equation II: Minimal mass dynamics, J. Eur. Math. Soc. 17 (2015), pp. 1855–1925. doi: 10.4171/JEMS/547
   Web of Science ®Google Scholar
• Y. Martel, F. Merle, and V. Raphaël, Blow up for the critical gKdV equation III: Exotic regimes, Ann. Sc. Norm. Sup. Pisa 14 (2015), pp. 575–631.
   Google Scholar
• D. Moldabayev, H. Kalisch, and D. Dutykh, The Whitham equation as a model for surface water waves, Phys. D 309 (2015), pp. 99–107. doi: 10.1016/j.physd.2015.07.010
   Web of Science ®Google Scholar
• D. Pelinovsky, Spectral stability of nonlinear waves in KdV-type evolution equations, in Spectral Analysis, Stability, Bifurcation in Modern Nonlinear Physical Systems, O.N. Kirilov and D.E. Pelinovsky, eds., Mechanical Engineering and Solid Mechanics Series, Wiley-ISTE Ltd., London, 2013, pp. 377–398.
   Google Scholar
• D.E. Pelinovsky and Y.A. Stepanyants, Convergence of Petviashvili's iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM J. Numer. Anal. 42 (2004), pp. 1110–1127. doi: 10.1137/S0036142902414232
   Web of Science ®Google Scholar
• V.I. Petviashvili, Equation of an extraordinary soliton, Soviet J. Plasma Phys. 2 (1976), pp. 257–258.
   Google Scholar
• Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (1986), pp. 856–869. doi: 10.1137/0907058
   Web of Science ®Google Scholar
• J.C. Saut, Sur quelques généralisations de l' équation de KdV I, J. Math. Pures Appl. 58 (1979), pp. 21–61.
   Google Scholar
• A. Sidi, Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms, SIAM J. Numer. Anal. 23 (1986), pp. 197–209. doi: 10.1137/0723014
   Web of Science ®Google Scholar
• A. Sidi, Practical Extrapolation Methods, Theory and Applications, Cambridge University Press, New York, 2003.
   Google Scholar
• A. Sidi, W.F. Ford, and D.A. Smith, Acceleration of convergence of vector sequences, SIAM J. Numer. Anal. 23 (1986), pp. 178–196. doi: 10.1137/0723013
   Web of Science ®Google Scholar
• D.A. Smith, W.F. Ford, and A. Sidi, Extrapolation methods for vector sequences, SIAM Rev. 29 (1987), pp. 199–233. doi: 10.1137/1029042
   Web of Science ®Google Scholar
• M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39 (1986), pp. 51–67. doi: 10.1002/cpa.3160390103
   Web of Science ®Google Scholar
• M.I. Weinstein, Existence and dynamic stability of solitary-wave solutions of equations arising in long wave propagation, Commun. Partial Differ. Equ. 12 (1987), pp. 1133–1173. doi: 10.1080/03605308708820522
   Web of Science ®Google Scholar
• G.B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. A 299 (1967), pp. 6–25. doi: 10.1098/rspa.1967.0119
   Google Scholar