Monoatomic chain: modulational instability and exact traveling wave solutions
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Monoatomic chain: modulational instability and exact traveling wave solutions Eric Tala-Tebue1,a , Guy Roger Deffo2,b , Serge Bruno Yamgoue3,c , Aurélien Kenfack-Jiotsa4,d , Francois Beceau Pelap2,e 1 Unité de Recherche d’Automatique et d’Informatique Appliquée, IUT-FV of Bandjoun, University of
Dschang, BP 134, Bandjoun, Cameroon
2 Unité de Recherche de Mécanique et de Modélisation des Systèmes Physiques (UR-2MSP), Faculté des
Sciences, Université de Dschang, BP 69, Dschang, Cameroon
3 Department of Physics, Higher Teacher Training College Bambili, University of Bamenda, P.O. Box 39,
Bamenda, Cameroon
4 Nonlinear Physics and Complex Systems Group, Department of Physics, The Higher Teachers’ Training
College, The University of Yaounde I, P.O. Box 47, Yaoundé, Cameroon Received: 27 April 2020 / Accepted: 20 July 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The objective of this work is to study a monoatomic chain which is described by the Klein–Gordon model with first and second neighbors anharmonic interactions. We consider specifically the case where both of these interactions are of cubic–quartic type and show, through the rotating wave approximation, that envelop waves in our model are governed by a modified nonlinear Schrödinger equation. The modulational instability is investigated on the latter, with a particular attention on the impact of the second neighbors. It is then revealed that the second neighbors interaction increases the regions of instability in the space of parameters and, hence, expands the possibility of propagating solitary waves in the network. More importantly, we derive the exact solutions of our modified nonlinear Schrödinger equation and show the influence of the second neighbors on them. For example, the width of these solutions can be controlled by adjusting the parameters relative to the second neighbors. Numerical simulations are done in order to confirm the analytical studies. Equally, the methods used in this study can also be implemented on other nonlinear equations.
1 Introduction Nonlinear equations are equations that attract the mind of several researchers nowadays. The peculiarity of this kind of equations is that they are used to described many complex phenomena or systems in diverse fields such as economy, biology, engineering, mathematics and physics. Particularly in physics, there is a great variety of phenomena governed by nonlinear equations. For instance, a dilute Bose Einstein condensates (BECs) with both the
a e-mail: [email protected] (corresponding author) b e-mail: [email protected] (corresponding author) c e-mail: [email protected] (corresponding author) d e-mail: [email protected] (corresponding author) e e-mail: [email protected] (corresponding author)
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two-body interactions of the condensate and either the feeding or the loss of atoms can be modeled, in the quasi-one-dimensional regime, by the inhomogeneou
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