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Stability of Solitary-Wave Solutions of Systems of Dispersive Equations

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Abstract

The present study is concerned with systems

$$\begin{aligned} \left\{ \begin{array}{ll} &{} \frac{\partial u}{\partial t} +\frac{\partial ^3 u}{\partial x^3} + \frac{\partial }{\partial x}P(u, v)=0,\\ &{} \frac{\partial v}{\partial t} +\frac{\partial ^3 v}{\partial x^3} + \frac{\partial }{\partial x}Q(u, v)=0, \end{array}\right. \end{aligned}$$

of Korteweg–de Vries type, coupled through their nonlinear terms. Here, \(u = u(x,t)\) and \(v = v(x,t)\) are real-valued functions of a real spatial variable x and a real temporal variable t. The nonlinearities P and Q are homogeneous, quadratic polynomials with real coefficients \(A,B,\ldots \), viz.

$$\begin{aligned} P(u,v)=Au^2+Buv+Cv^2, \qquad Q(u,v)=Du^2+Euv+Fv^2, \end{aligned}$$

in the dependent variables u and v. A satisfactory theory of local well-posedness is in place for such systems. Here, attention is drawn to their solitary-wave solutions. Special traveling waves termed proportional solitary waves are introduced and determined. Under the same conditions developed earlier for global well-posedness, stability criteria are obtained for these special, traveling-wave solutions.

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Acknowledgments

Helpful remarks by an anonymous referee improved the presentation. The three authors wish to record their thanks to the Archimedes Center for Modeling, Analysis and Computation at the University of Crete for support and hospitality during an important stage of this research. JB gratefully acknowledges support from the University of Illinois at Chicago and the Université de Paris Nord during part of this collaboration. HC received support from the Université Paris Val de Marne and OK was supported by the US National Science Foundation Grant DMS-1216740. JB and HC would like to thank the National Center for Theoretical Sciences, Mathematics Division, Taipei Office, for its support and hospitality. Helpful discussions of this work between the three authors took place at a workshop held at the American Institute of Mathematics (AIM, Palo Alto, CA). Thanks go to AIM for its organization and atmosphere which are so conducive to collaboration. The manuscript was finished while JB and HC were visiting the Department of Mathematics of the Ulsan National Institute of Science and Technology in South Korea.

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Bona, J.L., Chen, H. & Karakashian, O. Stability of Solitary-Wave Solutions of Systems of Dispersive Equations. Appl Math Optim 75, 27–53 (2017). https://doi.org/10.1007/s00245-015-9322-4

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