Geodesic
Periodic oscillations and waves in nonlinear weakly coupled dispersive systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems and methods in mechanics, Tome 300 (2018), pp. 158-167.

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Bifurcations of periodic solutions in autonomous nonlinear systems of weakly coupled equations are studied. A comparative analysis is carried out between the mechanisms of Lyapunov–Schmidt reduction of bifurcation equations for solutions close to harmonic oscillations and cnoidal waves. Sufficient conditions for the branching of orbits of solutions are formulated in terms of the Pontryagin functional depending on perturbing terms. 
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N. I. Makarenko; Z. V. Makridin. Periodic oscillations and waves in nonlinear weakly coupled dispersive systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems and methods in mechanics, Tome 300 (2018), pp. 158-167. http://geodesic.mathdoc.fr/item/TM_2018_300_a11/

[1] A. D. Bryuno, “The normal form of a system close to a Hamiltonian system”, Math. Notes, 48 (1990), 1100–1108 | DOI | MR | Zbl

[2] Gear J. A., Grimshaw R., “Weak and strong interactions between internal solitary waves”, Stud. Appl. Math., 70:3 (1984), 235–258 | DOI | MR

[3] A. T. Il'ichev, Solitary Waves in Hydromechanical Models, Fizmatlit, Moscow, 2003 (in Russian)

[4] L. G. Kurakin, V. I. Yudovich, “Application of the Lyapunov–Schmidt method to the problem of the branching of a cycle from a family of equilibria in a system with multicosymmetry”, Sib. Math. J., 41 (2000), 114–124 | DOI | MR | Zbl

[5] Littman W., “On the existence of periodic waves near critical speed”, Commun. Pure Appl. Math., 10:2 (1957), 241–269 | DOI | MR

[6] B. V. Loginov, “Branching Theory for the Solutions of Nonlinear Equations under Conditions of Group Invariance”, Fan, Tashkent, 1985 (in Russian) | MR

[7] B. V. Loginov, I. V. Konopleva, Yu. B. Rusak, “Symmetry and potentiality in the general problem of the branching theory”, Russ. Math., 50:4 (2006), 28–38 | MR | Zbl

[8] B. V. Loginov, V. A. Trenogin, “The use of group properties to determine multi-parameter families of solutions of nonlinear equations”, Math. USSR Sb., 14:3 (1971), 438–452 | DOI | MR | Zbl

[9] N. I. Makarenko, “On the bifurcation of solutions to invariant variational equations”, Dokl. Math., 53:3 (1996), 369–371 | MR

[10] I. G. Malkin, The Methods of Lyapunov and Poincaré in the Theory of Nonlinear Oscillations, URSS, Moscow, 2004 (in Russian) | MR

[11] V. K. Mel'nikov, “On the stability of the center for time-periodic perturbations”, Trans. Moscow Math. Soc., 12, 1963, 1–56 | MR | Zbl

[12] Mielke A., Hamiltonian and Lagrangian flows on center manifolds: With applications to elliptic variational problems, Lect. Notes Math., 1489, Springer, Berlin, 1991 | DOI | MR

[13] L. S. Pontryagin, “On dynamical systems close to Hamiltonian systems”, Zh. Eksp. Teor. Fiz., 4:9 (1934), 883–885

[14] A. M. Ter-Krikorov, “The existence of periodic waves which degenerate into a solitary wave”, J. Appl. Math. Mech., 24 (1961), 930–949 | DOI

[15] M. M. Vainberg, V. A. Trenogin, Theory of branching of solutions of non-linear equations, Noordhoff Int. Publ., Leyden, 1974 | MR | MR

[16] Vanderbauwhede A., “Lyapunov–Schmidt method for dynamical systems”, Encyclopedia of complexity and systems science, Springer, New York, 2009, 5299–5315 | DOI | MR

[17] Wright J. D., Scheel A., “Solitary waves and their linear stability in weakly coupled KdV equations”, Z. angew. Math. Phys., 58:4 (2007), 535–570 | DOI | MR

[18] V. I. Yudovich, “Cosymmetry, degeneration of solutions of operator equations, and onset of a filtration convection”, Math. Notes, 49 (1991), 540–545 | DOI | MR | Zbl

[19] V. I. Yudovich, “Implicit function theorem for cosymmetric equations”, Math. Notes, 60 (1996), 235–238 | DOI | DOI | MR | Zbl