Abstract
The analysis of stability of heteroclinic solutions to the Korteweg–de Vries–Burgers equation is generalized to the case of an arbitrary potential that gives rise to heteroclinic states. An example of a specific nonconvex potential is given for which there exists a wide set of heteroclinic solutions of different types. Stability of the corresponding solutions in the context of uniqueness of a solution to the problem of decay of an arbitrary discontinuity is discussed.
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References
J. C. Alexander and R. Sachs, “Linear instability of solitary waves of a Boussinesq type equation: A computer assisted computation,” Nonlinear World 2(4), 471–507 (1995).
A. P. Chugainova, “Nonstationary Solutions of a Generalized Korteweg–de Vries–Burgers Equation,” Tr. Mat. Inst. im._V.A. Steklova, Ross. Akad. Nauk 281, 215–223 (2013) [Proc. Steklov Inst. Math. 281, 204–212 (2013)].
A. P. Chugainova, A. T. Il’ichev, A. G. Kulikovskii, and V. A. Shargatov, “Problem of an arbitrary Selection conditions for thdiscontinuity disintegration for the generalized Hopf equation: e unique solution,” IMA J. Appl. Math. (in press).
A. P. Chugainova and V. A. Shargatov, “Stability of nonstationary solutions of the generalized KdV–Burgers equation,” Zh. Vychisl. Mat. Mat. Fiz. 55(2), 253–266 (2015) [Comput. Math. Math. Phys. 55, 251–263 (2015)].
A. P. Chugainova and V. A. Shargatov, “Stability of discontinuity structures described by a generalized KdV–Burgers equation,” Zh. Vychisl. Mat. Mat. Fiz. 56(2), 259–274 (2016) [Comput. Math. Math. Phys. 56, 263–277 (2016)].
I. M. Gel’fand, “Some problems in the theory of quasilinear equations,” Usp. Mat. Nauk 14(2), 87–158 (1959) [Am. Math. Soc. Transl., Ser. 2, 29, 295–381 (1963)].
S. K. Godunov, “On nonunique ‘blurring’ of discontinuities in solutions of quasilinear systems,” Dokl. Akad. Nauk SSSR 136(2), 272–273 (1961) [Sov. Math., Dokl. 2, 43–44 (1961)].
S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Kluwer, New York, 2003).
A. T. Il’ichev, “Envelope solitary waves and dark solitons at a water–ice interface,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 289, 163–177 (2015) [Proc. Steklov Inst. Math. 289, 152–166 (2015)].
A. T. Il’ichev, “Soliton-like structures on a water–ice interface,” Usp. Mat. Nauk 70(6), 85–138 (2015) [Russ. Math. Surv. 70, 1051–1103 (2015)].
A. T. Il’ichev, “Solitary wave packets beneath a compressed ice cover,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 32–42 (2016) [Fluid Dyn. 51(3), 327–337 (2016)].
A. T. Il’ichev, A. P. Chugainova, and V. A. Shargatov, “Spectral stability of special discontinuities,” Dokl. Akad. Nauk 462(5), 512–516 (2015) [Dokl. Math. 91(3), 347–351 (2015)].
A. T. Il’ichev and Y. B. Fu, “Stability of an inflated hyperelastic membrane tube with localized wall thinning,” Int. J. Eng. Sci. 80, 53–61 (2014).
A. G. Kulikovskii, “A possible effect of oscillations in the structure of a discontinuity on the set of admissible discontinuities,” Dokl. Akad. Nauk SSSR 275(6), 1349–1352 (1984) [Sov. Phys., Dokl. 29(4), 283–285 (1984)].
A. G. Kulikovskii and A. P. Chugainova, “Simulation of the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena,” Zh. Vychisl. Mat. Mat. Fiz. 44(6), 1119–1126 (2004) [Comput. Math. Math. Phys. 44, 1062–1068 (2004)].
A. G. Kulikovskii and A. P. Chugainova, “Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory,” Usp. Mat. Nauk 63(2), 85–152 (2008) [Russ. Math. Surv. 63, 283–350 (2008)].
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Chapman & Hall/CRC, Boca Raton, FL, 2001), Monogr. Surv. Pure Appl. Math.118.
R. L. Pego, P. Smereka, and M. I. Weinstein, “Oscillatory instability of traveling waves for a KdV–Burgers equation,” Physica D 67(1–3), 45–65 (1993).
R. L. Pego and M. I. Weinstein, “Eigenvalues, and instabilities of solitary waves,” Philos. Trans. R. Soc. London A 340(1656), 47–94 (1992).
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 163–173.
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Il’ichev, A.T., Chugainova, A.P. Spectral stability theory of heteroclinic solutions to the Korteweg-de Vries-Burgers equation with an arbitrary potential. Proc. Steklov Inst. Math. 295, 148–157 (2016). https://doi.org/10.1134/S0081543816080083
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DOI: https://doi.org/10.1134/S0081543816080083