# American Institute of Mathematical Sciences

August  2016, 9(4): 1025-1038. doi: 10.3934/dcdss.2016040

## Multiple homoclinic solutions for a one-dimensional Schrödinger equation

 1 Department of Mathematics - University of Torino, Via Carlo Alberto, 10 - 10123 Torino 2 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, via delle Scienze 206, 33100 Udine, Italy

Received  July 2015 Revised  January 2016 Published  August 2016

In this paper we study the problem of the existence of homoclinic solutions to a Schrödinger equation of the form $x''-V(t)x+x^3=0,$ where is a stepwise potential. The technique of proof is based on a topological method, relying on the properties of the transformation of continuous planar paths (the S.A.P. method), together with the application of the classical Conley-Ważewski's method.
Citation: Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040
##### References:
 [1] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067. [2] A. Boscaggin, W. Dambrosio and D. Papini, Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation, Nonlinearity, 28 (2015), 3465-3485. doi: 10.1088/0951-7715/28/10/3465. [3] A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438. doi: 10.1006/jdeq.2001.4080. [4] C. Conley, An application of Wa.zewski's method to a non-linear boundary value problem which arises in population genetics, J. Math. Biol., 2 (1975), 241-249. doi: 10.1007/BF00277153. [5] M. del Pino, P. Felmer and O. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential, Nonlinear Anal., 34 (1998), 979-989. doi: 10.1016/S0362-546X(97)00593-2. [6] E. Ellero and F. Zanolin, Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: multiplicity results for stepwise potentials, Bound. Value Probl., 2013 (2013), 23pp. doi: 10.1186/1687-2770-2013-167. [7] E. Ellero and F. Zanolin, Connected branches of initial points for asymptotic solutions with applications, in preparation. [8] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0. [9] A. Gavioli, Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 79-100. doi: 10.1007/s00030-010-0085-y. [10] P. J. Holmes and C. A. Stuart, Homoclinic orbits for eventually autonomous planar flows, Z. Angew. Math. Phys., 43 (1992), 598-625. doi: 10.1007/BF00946253. [11] K. Kuratowski, Topology, Vol. 2, Academic Press, New York, 1968. [12] A. Margheri, C. Rebelo and F. Zanolin, Connected branches of initial points for asymptotic BVPs, with application to heteroclinic and homoclinic solutions, Adv. Nonlinear Stud., 9 (2009), 95-135. [13] J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Differential Equations, 188 (2003), 33-51. doi: 10.1016/S0022-0396(02)00073-6. [14] J. S. Muldowney and D. Willett, An elementary proof of the existence of solutions to second order nonlinear boundary value problems, SIAM J. Math. Anal., 5 (1974), 701-707. doi: 10.1137/0505068. [15] D. Papini and F. Zanolin, A topological approach to superlinear indefinite boundary value problems, Topol. Methods Nonlinear Anal., 15 (2000), 203-233. [16] D. Papini and F. Zanolin, Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells, Fixed Point Theory Appl., 2004, 113-134. doi: 10.1155/S1687182004401028. [17] D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91. [18] A. Pascoletti, M. Pireddu and F. Zanolin, Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, The 8th Colloquium on the Qualitative Theory of Differential Equations, No. 14, 32 pp., Proc. Colloq. Qual. Theory Differ. Equ., 8, Electron. J. Qual. Theory Differ. Equ., Szeged, 2008. [19] M. Pireddu and F. Zanolin, Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on N-dimensional cells, Adv. Nonlinear Stud., 5 (2005), 411-440. [20] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [21] C. Rebelo and F. Zanolin, On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm-Liouville problems via the shooting map, Differential Integral Equations, 13 (2000), 1473-1502. [22] M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order, J. Differential Equations, 37 (1980), 285-295. doi: 10.1016/0022-0396(80)90099-6. [23] T. Wa.zewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math., 20 (1947), 279-313 (1948). [24] C. Zanini and F. Zanolin, Complex dynamics in one-dimensional nonlinear Schrödinger equations with stepwise potential, preprint, 2014.

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##### References:
 [1] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067. [2] A. Boscaggin, W. Dambrosio and D. Papini, Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation, Nonlinearity, 28 (2015), 3465-3485. doi: 10.1088/0951-7715/28/10/3465. [3] A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438. doi: 10.1006/jdeq.2001.4080. [4] C. Conley, An application of Wa.zewski's method to a non-linear boundary value problem which arises in population genetics, J. Math. Biol., 2 (1975), 241-249. doi: 10.1007/BF00277153. [5] M. del Pino, P. Felmer and O. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential, Nonlinear Anal., 34 (1998), 979-989. doi: 10.1016/S0362-546X(97)00593-2. [6] E. Ellero and F. Zanolin, Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: multiplicity results for stepwise potentials, Bound. Value Probl., 2013 (2013), 23pp. doi: 10.1186/1687-2770-2013-167. [7] E. Ellero and F. Zanolin, Connected branches of initial points for asymptotic solutions with applications, in preparation. [8] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0. [9] A. Gavioli, Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 79-100. doi: 10.1007/s00030-010-0085-y. [10] P. J. Holmes and C. A. Stuart, Homoclinic orbits for eventually autonomous planar flows, Z. Angew. Math. Phys., 43 (1992), 598-625. doi: 10.1007/BF00946253. [11] K. Kuratowski, Topology, Vol. 2, Academic Press, New York, 1968. [12] A. Margheri, C. Rebelo and F. Zanolin, Connected branches of initial points for asymptotic BVPs, with application to heteroclinic and homoclinic solutions, Adv. Nonlinear Stud., 9 (2009), 95-135. [13] J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Differential Equations, 188 (2003), 33-51. doi: 10.1016/S0022-0396(02)00073-6. [14] J. S. Muldowney and D. Willett, An elementary proof of the existence of solutions to second order nonlinear boundary value problems, SIAM J. Math. Anal., 5 (1974), 701-707. doi: 10.1137/0505068. [15] D. Papini and F. Zanolin, A topological approach to superlinear indefinite boundary value problems, Topol. Methods Nonlinear Anal., 15 (2000), 203-233. [16] D. Papini and F. Zanolin, Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells, Fixed Point Theory Appl., 2004, 113-134. doi: 10.1155/S1687182004401028. [17] D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91. [18] A. Pascoletti, M. Pireddu and F. Zanolin, Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, The 8th Colloquium on the Qualitative Theory of Differential Equations, No. 14, 32 pp., Proc. Colloq. Qual. Theory Differ. Equ., 8, Electron. J. Qual. Theory Differ. Equ., Szeged, 2008. [19] M. Pireddu and F. Zanolin, Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on N-dimensional cells, Adv. Nonlinear Stud., 5 (2005), 411-440. [20] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [21] C. Rebelo and F. Zanolin, On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm-Liouville problems via the shooting map, Differential Integral Equations, 13 (2000), 1473-1502. [22] M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order, J. Differential Equations, 37 (1980), 285-295. doi: 10.1016/0022-0396(80)90099-6. [23] T. Wa.zewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math., 20 (1947), 279-313 (1948). [24] C. Zanini and F. Zanolin, Complex dynamics in one-dimensional nonlinear Schrödinger equations with stepwise potential, preprint, 2014.
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