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
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern Methods of Mechanics, Tome 322 (2023), pp. 266-281

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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{TM_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 = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {266--281},
     publisher = {mathdoc},
     volume = {322},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2023_322_a20/}
}
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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. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern Methods of Mechanics, Tome 322 (2023), pp. 266-281. http://geodesic.mathdoc.fr/item/TM_2023_322_a20/