Geodesic
Structures of Classical and Special Discontinuities for the Generalized Korteweg--de Vries--Burgers Equation in the Case of a Flux Function with Four Inflection Points
Informatics and Automation, Modern Methods of Mechanics, Tome 322 (2023), pp. 266-281.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the structure of the set of traveling wave solutions for the generalized Korteweg–de Vries–Burgers equation with the flux function having four inflection points. In this case there arise two monotone structures of stable special discontinuities propagating at different velocities (such a situation has not been described earlier in the literature). Both structures of special discontinuities are linearly stable. To analyze the linear stability of the structures of classical and special discontinuities, we apply a method based on the use of the Evans function. We also propose a conjecture that establishes the admissibility of classical discontinuities in the case when there are two stable special discontinuities.
Keywords: Hopf equation, Korteweg–de Vries–Burgers equation, singular discontinuities, Evans function. 
@article{TRSPY_2023_322_a20,
     author = {V. A. Shargatov and A. P. Chugainova and A. M. Tomasheva},
     title = {Structures of {Classical} and {Special} {Discontinuities} for the {Generalized} {Korteweg--de} {Vries--Burgers} {Equation} in the {Case} of a {Flux} {Function} with {Four} {Inflection} {Points}},
     journal = {Informatics and Automation},
     pages = {266--281},
     publisher = {mathdoc},
     volume = {322},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2023_322_a20/}
}
TY  - JOUR
AU  - V. A. Shargatov
AU  - A. P. Chugainova
AU  - A. M. Tomasheva
TI  - Structures of Classical and Special Discontinuities for the Generalized Korteweg--de Vries--Burgers Equation in the Case of a Flux Function with Four Inflection Points
JO  - Informatics and Automation
PY  - 2023
SP  - 266
EP  - 281
VL  - 322
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2023_322_a20/
LA  - ru
ID  - TRSPY_2023_322_a20
ER  - 
%0 Journal Article
%A V. A. Shargatov
%A A. P. Chugainova
%A A. M. Tomasheva
%T Structures of Classical and Special Discontinuities for the Generalized Korteweg--de Vries--Burgers Equation in the Case of a Flux Function with Four Inflection Points
%J Informatics and Automation
%D 2023
%P 266-281
%V 322
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2023_322_a20/
%G ru
%F TRSPY_2023_322_a20
V. A. Shargatov; A. P. Chugainova; A. M. Tomasheva. Structures of Classical and Special Discontinuities for the Generalized Korteweg--de Vries--Burgers Equation in the Case of a Flux Function with Four Inflection Points. Informatics and Automation, Modern Methods of Mechanics, Tome 322 (2023), pp. 266-281. http://geodesic.mathdoc.fr/item/TRSPY_2023_322_a20/

[1] I. B. Bakholdin, “The structure of evolutional jumps in reversible systems”, J. Appl. Math. Mech., 63:1 (1999), 45–53 | DOI | MR | Zbl

[2] I. B. Bakholdin, “Jumps in models described by generalized Korteweg–de Vries equations”, Fluid Dyn., 34:4 (1999), 534–545 | MR | Zbl

[3] Bakholdin I.B., Bezdissipativnye razryvy v mekhanike sploshnoi sredy, Fizmatlit, M., 2004

[4] I. B. Bakholdin, “Time-invariant and time-varying discontinuity structures for models described by the generalized Korteweg–Burgers equation”, J. Appl. Math. Mech., 75:2 (2011), 189–209 | DOI | MR | Zbl

[5] I. B. Bakholdin, “Theory and classification of the reversible structures of discontinuities in hydrodynamic-type models”, J. Appl. Math. Mech., 78:6 (2014), 599–612 | DOI | MR | Zbl

[6] Bona J.L., Schonbeck M.E., “Travelling-wave solutions to the Korteweg–de Vries–Burgers equation”, Proc. R. Soc. Edinburgh A, 101:3–4 (1985), 207–226 | DOI | MR | Zbl

[7] Chugainova A.P., Il'ichev A.T., Kulikovskii A.G., Shargatov V.A., “Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: Selection conditions for a unique solution”, IMA J. Appl. Math., 82:3 (2017), 496–525 | MR | Zbl

[8] Chugainova A.P., Kolomiytsev G.V., Shargatov V.A., “On the instability of monotone traveling-wave solutions for a generalized Korteweg–de Vries–Burgers equation”, Russ. J. Math. Phys., 29:3 (2022), 342–357 | DOI | MR | Zbl

[9] A. P. Chugainova and V. A. Shargatov, “Stability of nonstationary solutions of the generalized KdV–Burgers equation”, Comput. Math. Math. Phys., 55:2 (2015), 251–263 | DOI | MR | Zbl

[10] Chugainova A.P., Shargatov V.A., “Traveling waves and undercompressive shocks in solutions of the generalized Korteweg–de Vries–Burgers equation with a time-dependent dissipation coefficient distribution”, Eur. Phys. J. Plus, 135:8 (2020), 635 | DOI

[11] El G.A., Hoefer M.A., Shearer M., “Dispersive and diffusive–dispersive shock waves for nonconvex conservation laws”, SIAM Rev., 59:1 (2017), 3–61 | DOI | MR | Zbl

[12] I. M. Gel'fand, “Some problems in the theory of quasilinear equations”, Am. Math. Soc. Transl., Ser. 2, 29 (1963), 295–381 | MR | MR | Zbl | Zbl

[13] Grad H., Hu P.N., “Unified shock profile in a plasma”, Phys. Fluids, 10:12 (1967), 2596–2602 | DOI

[14] Hayes B., Shearer M., “Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes”, Proc. R. Soc. Edinburgh A, 129:4 (1999), 733–754 | DOI | MR | Zbl

[15] Hayes B.T., Shearer M., “A nonconvex scalar conservation law with trilinear flux”, Q. Appl. Math., 59:4 (2001), 615–635 | DOI | MR | Zbl

[16] A. P. Il'Ichev and A. T. Chugainova, “Spectral stability theory of heteroclinic solutions to the Korteweg–de Vries–Burgers equation with an arbitrary potential”, Proc. Steklov Inst. Math., 295 (2016), 148–157 | DOI | DOI | MR | Zbl

[17] A. T. Il'ichev, A. P. Chugainova, and V. A. Shargatov, “Spectral stability of special discontinuities”, Dokl. Math., 91:3 (2015), 347–351 | DOI | MR | Zbl

[18] Jacobs D., McKinney B., Shearer M., “Travelling wave solutions of the modified Korteweg–de Vries–Burgers equation”, J. Diff. Eqns., 116:2 (1995), 448–467 | DOI | MR | Zbl

[19] Johnson R.S., “A non-linear equation incorporating damping and dispersion”, J. Fluid Mech., 42:1 (1970), 49–60 | DOI | MR | Zbl

[20] A. G. Kulikovskii, “A possible effect of oscillations in the structure of a discontinuity on the set of admissible discontinuities”, Sov. Phys., Dokl., 29:4 (1984), 283–285 | MR

[21] A. G. Kulikovskii, “Strong discontinuities in flows of continuous media, and their structure”, Proc. Steklov Inst. Math., 182 (1990), 285–317 | MR

[22] 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”, Comput. Math. Math. Phys., 44:6 (2004), 1062–1068 | MR | Zbl

[23] A. G. Kulikovskii and A. P. Chugainova, “Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory”, Russ. Math. Surv., 63:2 (2008), 283–350 | DOI | DOI | MR | Zbl

[24] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 118, Chapman Hall/CRC, Boca Raton, FL, 2001 | MR | Zbl

[25] Lax P.D., “Hyperbolic systems of conservation laws. II”, Commun. Pure Appl. Math., 10:4 (1957), 537–566 | DOI | MR | Zbl

[26] LeFloch P.G., Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves, Birkhäuser, Basel, 2002 | MR | Zbl

[27] LeFloch P.G., Shearer M., “Nonclassical Riemann solvers with nucleation”, Proc. R. Soc. Edinburgh A, 134:5 (2004), 961–984 | DOI | MR | Zbl

[28] Lyapidevskii V.Yu., Teshukov V.M., Matematicheskie modeli rasprostraneniya dlinnykh voln v neodnorodnoi zhidkosti, Izd-vo SO RAN, Novosibirsk, 2000

[29] Pego R.L., Smereka P., Weinstein M.I., “Oscillatory instability of traveling waves for a KdV–Burgers equation”, Physica D, 67:1–3 (1993), 45–65 | DOI | MR | Zbl

[30] Pego R.L., Weinstein M.I., “Eigenvalues, and instabilities of solitary waves”, Philos. Trans. R. Soc. London A, 340:1656 (1992), 47–94 | DOI | MR | Zbl

[31] B. L. Rozhdestvenskii, “Discontinuous solutions of hyperbolic systems of quasilinear equations”, Russ. Math. Surv., 15:6 (1960), 53–111 | DOI | MR | Zbl

[32] A. V. Samokhin, “Reflection and refraction of solitons by the KdV–Burgers equation in nonhomogeneous dissipative media”, Theor. Math. Phys., 197:1 (2018), 1527–1533 | DOI | DOI | MR | Zbl

[33] Samokhin A., “The KdV soliton crosses a dissipative and dispersive border”, Diff. Geom. Appl., 75 (2021), 101723 | DOI | MR | Zbl

[34] Shargatov V.A., Chugainova A.P., “Stability analysis of traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation with variable dissipation parameter”, J. Comput. Appl. Math., 397 (2021), 113654 | DOI | MR | Zbl

[35] Shargatov V.A., Chugainova A.P., Kolomiytsev G.V., “Global stability of traveling wave solutions of generalized Korteweg–de Vries–Burgers equation with non-constant dissipation parameter”, J. Comput. Appl. Math., 412 (2022), 114354 | DOI | MR | Zbl