Dias, João-Paulo; Freitas, Pedro Decay of solutions for a class of nonlinear Schrödinger equations in \(\mathbb{R}\) and the stability of shock profiles for a quasilinear Benney system. (English) Zbl 1390.35322 Nonlinearity 31, No. 3, 1110-1119 (2018). Summary: We study the stability in a partial linearisation framework of a particular travelling wave \((0,\tilde{V})\) of the quasilinear Benney system, where \( \tilde{V}\) is a standing wave with a shock profile for the viscous conservation law. Cited in 1 Document MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 35C07 Traveling wave solutions 35C20 Asymptotic expansions of solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 76L05 Shock waves and blast waves in fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction Keywords:Benney system; standing wave; stability; nonlinear Schrödinger equation PDFBibTeX XMLCite \textit{J.-P. Dias} and \textit{P. Freitas}, Nonlinearity 31, No. 3, 1110--1119 (2018; Zbl 1390.35322) Full Text: DOI References: [1] Benney, D. J., A general theory for interactions between short and long waves, Stud. Appl. Math., 56, 81-94, (1977) · Zbl 0358.76011 · doi:10.1002/sapm197756181 [2] Cazenave, T., Semilinear Schrödinger Equations, (2003), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1055.35003 [3] Cuccagna, S.; Georgiev, V.; Visciglia, N., Decay and scattering of small solutions of pure power NLS in \(\newcommand{\R}{\mathbb{R}} \R\) with {\it p} > 3 and with a potential, Commun. Pure Appl. Math., 67, 957-981, (2014) · Zbl 1293.35290 · doi:10.1002/cpa.21465 [4] Dias, J. P.; Figueira, M., Existence of weak solutions for a quasilinear version of Benney equations, J. Hyperb. Differ. Equ., 4, 555-563, (2007) · Zbl 1148.35047 · doi:10.1142/S0219891607001252 [5] Dias, J. P.; Figueira, M.; Oliveira, F., Existence of local strong solutions for a quasilinear Benney system, C. R. Acad. Sci. Paris I, 344, 493-496, (2007) · Zbl 1114.35002 · doi:10.1016/j.crma.2007.03.005 [6] Dias, J. P.; Figueira, M.; Frid, H., Vanishing viscosity with short wave-long wave interactions for systems of conservative laws, Arch. Ration. Mech. Anal., 196, 981-1010, (2010) · Zbl 1203.35020 · doi:10.1007/s00205-009-0273-2 [7] Il’in, A. M.; Oleinik, O. A., Asymptotic behaviour of the solutions of the Cauchy problem for certain quasilinear equations for large time (Russian), Math. Sb., 51, 191-216, (1960) [8] Kawashima, S.; Matsumura, A., Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101, 97-127, (1985) · Zbl 0624.76095 · doi:10.1007/BF01212358 [9] Lions, J. L.; Magenes, E., Problèmes Aux Limites Non Homogènes et Applications, vol 1, (1968), Paris: Dunod, Paris · Zbl 0165.10801 [10] Naumkin, I. P., Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential, J. Math. Phys., 57, (2016) · Zbl 1343.81206 · doi:10.1063/1.4948743 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.