Non-analytic solutions of nonlinear wave models

Nonlinear Theory of Generalized Functions

Volumn , Issue , 2022, Pages 129-145

  Li, Yi A ^a   Olver, Peter J ^a   Rosenau, Philip ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85136519936     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (29)

1. Beals, R., Sattinger, D.H., Szmigielski, J., Acoustic scattering and the extended Korteweg-de Vries hierarchy, preprint, University of Minnesota, 1998.
2. Benjamin, T.B., Bona, J.L., Mahoney, J.J., Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. London A 272 (1972), 47-78.
3. Bona, J.L., Pritchard, W.G., Scott, L.R., Solitary-wave interaction, Phys. Fluids 23 (1980), 438-441.
4. Boussinesq, J., Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (2) (1872), 55-108.
5. Boussinesq, J., Essai sur la théorie des eaux courants, Mém. Acad. Sei. Inst. Nat. France, 23 (1) (1877), 1-680.
6. Camassa, R., Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664.
7. Camassa, R., Holm, D.D., Hyman, J.M., A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1-33.
8. Constantin, A., McKean, H., The integrability of a shallow water equation with periodic boundary conditions, preprint, Courant Institute, 1997.
9. Drazin, P.G., Johnson, R.S., Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.
10. Fokas, A.S., Olver, P.J., Rosenau, P., A plethora of integrable bi-Hamiltonian equations, Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, A.S. Fokas and I.M. Gel'fand, eds., Progress in Nonlinear Differential Equations vol. 26, Birkhäuser, Boston, 1996, 93-101.
11. Fuchssteiner, B., The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional abelian symmetry groups, Prog. Theor. Phys. 65 (1981) 861-876.
12. Gardner, L.R.T., Gardner, G.A., Solitary waves of the regularised long wave equation, J. Comp. Phys. 91 (1990), 441-459.
13. Kaup, D.J., A higher-order water-wave equation and the method for solving it, Prog. Theor. Physics 54 (1975), 396-408.
14. Korteweg, D.J., de Vries, G., On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag.; (5) 39 (1895), 422-443.
15. Kupershmidt, B.A., Mathematics of dispersive waves, Comm. Math. Phys.; 99 (1985), 51-73.
16. Li, Y.A., Weak solutions of a generalized Boussinesq system, J. Dyn. Diff. Eq., to appear.
17. LI, Y.A., Olver, P.J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. I. Compactons and peakons, Discrete Cont. Dyn. Syst. 3 (1997), 419-432.
18. Li, Y.A., Olver, P. J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. II. Complex analytic behavior and convergence to non-analytic solutions; Discrete Cont. Dyn. Syst. 4 (1998), 159.
19. Makhankov, V.G., Dynamics of classical solitons (in non-integrable systems), Phys. Rep. 35 (1978), 1-128.
20. Olver, P.J., Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993.
21. Olver, P.J., Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E; 53 (1996), 1900-1906.
22. Rosenau, P., Nonlinear dispersion and compact structures, Phys. Rev. Lett. 73 (1994), 1737-1740.
23. Rosenau, P., On nonanalytic waves formed by a nonlinear dispersion, Phys. Lett. A 230 (1997), 305-318.
24. Rosenau, P., Hyman, J.M., Compactons: solitons with finite wavelength, Phys. Rev. Lett. 70 (1993), 564-567.
25. Santarelli, A.R., Numerical analysis of the regularized long-wave equation: anelastic collision of solitary waves; II Nuovo Cim. 46B (1978), 179-188.
26. Scott Russell, J., Report on Water Waves, British Assoc. Report, 1844.
27. toland, J.F., On the existence of a wave of greatest height and Stokes' conjecture, Proc. Roy. Soc. London A 363 (1978), 469-485.
28. Wadati, M., Ichikawa, Y.H., Shimizu, T., Cusp soliton of a new integrable nonlinear evolution equation, Prog. Theor. Phys. 64 (1980), 1959-1967.
29. Whitham, G.B., Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.