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October  2016, 36(10): 5721-5741. doi: 10.3934/dcds.2016051

Solitary waves for an internal wave model

 1 Departamento de Matemáticas, Universidad del Valle, Calle 13 Nro. 100-00, Cali, Colombia, Colombia

Received  August 2015 Revised  April 2016 Published  July 2016

We show the existence of solitary wave solutions of finite energy for a model to describe the propagation of internal waves for wave speed $c$ large enough. Furthermore, some of these solutions are approximated using a Newton-type iteration combined with a collocation-spectral strategy for spatial discretization of the corresponding solitary wave equations.
Citation: José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051
References:
 [1] J. Angulo, M. Scialom and C. Banquet, The regularized Benjamin-Ono and BBM equations: Well possedness and nonlinear stability, J. Diff. Eq., 250 (2011), 4011-4036. doi: 10.1016/j.jde.2010.12.016. [2] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X. [3] T. B. Benjamin, J. L. Bona and D. K. Bose, Solitary-wave solutions of nonlinear problems, Phil. Trans. R. Soc. Lond. A, 331 (1990), 195-244. doi: 10.1098/rsta.1990.0065. [4] D. J. Benney and J. C. Luke, Interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), 309-313. doi: 10.1002/sapm1964431309. [5] R. Camassa and D. Holm, An integrable shallow water with peaked solitons, Phys. Rev. Lett. , 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [6] H. Chen, Existence of periodic travelling-wave solutions of nonlinear, dispersive wave equations, Nonlinearity, 17 (2004), 2041-2056. doi: 10.1088/0951-7715/17/6/003. [7] H. Chen, M. Chen and V. Nguyen, Cnoidal wave solutions to Boussinesq systems, Nonlinearity, 20 (2007), 1443-1461. doi: 10.1088/0951-7715/20/6/007. [8] J. Conway, Functions of one Complex Variable, 2ed., Graduate Texts in Mathematics, Springer, New York, 1978. [9] A. Degasperis and M. Procesi, Asymptotic integrability, in symmetry and Perturbation Theory(A. Degasperis and G. Gaeta, eds.), World Scientific Publishing, 23-37, 1999. [10] J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. [11] A. Granas, The Leray-Schauder index and fixed point theory for arbitrary ANRs, Bull. Soc. Math. Fr., 100 (1972), 209-228. [12] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739. [13] M. A. Krasnosel'skii, Positive Solutions of Operators Equations, ed. L.F. Boron: P. Noordhoff Ltd., 1964. [14] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, ed. J. Burlak, NY, USA: Pergamon Press, 1964. [15] Z. Linghai, Decay of solutions of generalized Benjamin-Bona-Mahony equations, Acta Mathematica Sinica, 10 (1994), 428-438. doi: 10.1007/BF02582039. [16] J. C. Muñoz Grajales, Existence and numerical approximation of solutions of an improved internal wave model, Math. Model. Anal. 19 (2014), 309-333. doi: 10.3846/13926292.2014.924039. [17] F. Pipicano and J. C. Muñoz Grajales, Existence of periodic travelling wave solutions for a regularized Benjamin-Ono system, J. Diff. Eq. 259 (2015), 7503-7528. doi: 10.1016/j.jde.2015.08.030. [18] J. Quintero and R. Pego, Two-dimensional solitary waves for a Benney-luke equation, Physica D, 132 (1999), 476-496. doi: 10.1016/S0167-2789(99)00058-5. [19] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. [20] The Bateman Manuscript Project, California Institute of Technology, Ed. A. Erdeélyi, McGraw-Hill, New York, 1954.

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References:
 [1] J. Angulo, M. Scialom and C. Banquet, The regularized Benjamin-Ono and BBM equations: Well possedness and nonlinear stability, J. Diff. Eq., 250 (2011), 4011-4036. doi: 10.1016/j.jde.2010.12.016. [2] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X. [3] T. B. Benjamin, J. L. Bona and D. K. Bose, Solitary-wave solutions of nonlinear problems, Phil. Trans. R. Soc. Lond. A, 331 (1990), 195-244. doi: 10.1098/rsta.1990.0065. [4] D. J. Benney and J. C. Luke, Interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), 309-313. doi: 10.1002/sapm1964431309. [5] R. Camassa and D. Holm, An integrable shallow water with peaked solitons, Phys. Rev. Lett. , 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [6] H. Chen, Existence of periodic travelling-wave solutions of nonlinear, dispersive wave equations, Nonlinearity, 17 (2004), 2041-2056. doi: 10.1088/0951-7715/17/6/003. [7] H. Chen, M. Chen and V. Nguyen, Cnoidal wave solutions to Boussinesq systems, Nonlinearity, 20 (2007), 1443-1461. doi: 10.1088/0951-7715/20/6/007. [8] J. Conway, Functions of one Complex Variable, 2ed., Graduate Texts in Mathematics, Springer, New York, 1978. [9] A. Degasperis and M. Procesi, Asymptotic integrability, in symmetry and Perturbation Theory(A. Degasperis and G. Gaeta, eds.), World Scientific Publishing, 23-37, 1999. [10] J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. [11] A. Granas, The Leray-Schauder index and fixed point theory for arbitrary ANRs, Bull. Soc. Math. Fr., 100 (1972), 209-228. [12] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739. [13] M. A. Krasnosel'skii, Positive Solutions of Operators Equations, ed. L.F. Boron: P. Noordhoff Ltd., 1964. [14] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, ed. J. Burlak, NY, USA: Pergamon Press, 1964. [15] Z. Linghai, Decay of solutions of generalized Benjamin-Bona-Mahony equations, Acta Mathematica Sinica, 10 (1994), 428-438. doi: 10.1007/BF02582039. [16] J. C. Muñoz Grajales, Existence and numerical approximation of solutions of an improved internal wave model, Math. Model. Anal. 19 (2014), 309-333. doi: 10.3846/13926292.2014.924039. [17] F. Pipicano and J. C. Muñoz Grajales, Existence of periodic travelling wave solutions for a regularized Benjamin-Ono system, J. Diff. Eq. 259 (2015), 7503-7528. doi: 10.1016/j.jde.2015.08.030. [18] J. Quintero and R. Pego, Two-dimensional solitary waves for a Benney-luke equation, Physica D, 132 (1999), 476-496. doi: 10.1016/S0167-2789(99)00058-5. [19] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. [20] The Bateman Manuscript Project, California Institute of Technology, Ed. A. Erdeélyi, McGraw-Hill, New York, 1954.
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