Stability Of Peakons For The Generalized Camassa-holm Equation

We study the existence of minimizers for a constrained variational problems in H1(ℝ). These minimizers are stable waves solutions for the Generalized Camassa-Holm equation, and their derivative may have a singularity (in which case the travelling wave is called a peakon). The existence result is based on a method developed by the same author in a previous work. By giving examples, we show how our method works.2003112Brezis, H., Lieb, E., Minimum Action Solutions of Some Vector Field Equations (1984) Comm. Math. Phys., 96 (1), pp. 97-113Cazenave, T., Lions, P.L., Orbital Stability of Standing waves for Some Nonlinear Schrodinger Equations (1982) Comm. Math. Phys., 85, pp. 549-561Constantin, A., Molinet, L., Orbital Stability of Solitary Waves for a Shallow Water Equation (2001) Physica D, 157, pp. 75-89Constantin, A., Strauss, W., Stability of Peakons (2000) Comm. Pure and Applied Math., 53, pp. 603-610Constantin, A., Strauss, W., Stability of the Camassa-Holm Solitons (2002) J. Nonlinear Science, 12, pp. 415-422Gradshteyn, I., Ryzhik, I., (1965) Tables of Integrals, Series and Products, 4th Edition, , Academic PressGrillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry I (1987) J. Funct. Anal., 74 (1), pp. 160-197Lions, P.L., The Concentration-Compactness Principle in the Calculus of Variations (1984) Ann. Inst. H. Poincaré, Anal. Non Linéaire, Sec A, 1 (1 PART), pp. 109-145Ann. Inst. H. Poincaré, Anal. Non Linéaire, Sec A, (2 PART), pp. 223-283Liu, Z., Qian, T., Peakons and their Bifurcation in a Generalized Camassa-Holm Equation (2001) Intern. J. of Bifurcation and Chaos, 11 (3), pp. 781-792Lopes, O., Nonlocal variational problems arising in long wave propagation (2000) Control, Optimisation and Calculus of Variations (COCV), 5, pp. 501-528. , http://www.emath.fr/coc