Geodesic
An effective algorithm for finding multidimensional conservation laws for integrable systems of hydrodynamic type
Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 2, pp. 320-330 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study a new property of integrable systems, the existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional hydrodynamic chains. We describe an effective algorithm for successively computing an infinite set of three-dimensional conservation laws for the Benney pair of commuting flows.
Keywords: integrable system, overdetermined system, hydrodynamic Benney chain, Kadomtsev–Petviashvili equation, three-dimensional conservation law. 
@article{TMF_2018_194_2_a6,
     author = {Z. V. Makridin},
     title = {An~effective algorithm for finding multidimensional conservation laws for integrable systems of hydrodynamic type},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {320--330},
     year = {2018},
     volume = {194},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_194_2_a6/}
}
TY  - JOUR
AU  - Z. V. Makridin
TI  - An effective algorithm for finding multidimensional conservation laws for integrable systems of hydrodynamic type
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 320
EP  - 330
VL  - 194
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_194_2_a6/
LA  - ru
ID  - TMF_2018_194_2_a6
ER  - 
%0 Journal Article
%A Z. V. Makridin
%T An effective algorithm for finding multidimensional conservation laws for integrable systems of hydrodynamic type
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 320-330
%V 194
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2018_194_2_a6/
%G ru
%F TMF_2018_194_2_a6
Z. V. Makridin. An effective algorithm for finding multidimensional conservation laws for integrable systems of hydrodynamic type. Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 2, pp. 320-330. http://geodesic.mathdoc.fr/item/TMF_2018_194_2_a6/

[1] I. M. Krichever, “Metod usredneniya Uizema dlya dvumernykh ‘integriruemykh’ uravnenii”, Funkts. analiz i ego pril., 22:3 (1988), 37–52 ; “Спектральная теория двумерных периодических операторов и ее приложения”, УМН, 44:2 (1989), 121–184 ; I. M. Krichever, “The dispersionless Lax equations and topological minimal models”, Commun. Math. Phys., 143:2 (1992), 415–429 ; “The $\tau$-function of the universal Whitham hierarchy, matrix models and topological field theories”, Commun. Pure Appl. Math., 47:4 (1994), 437–475 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[2] E. V. Ferapontov, K. R. Khusnutdinova, “On integrability of $(2+1)$-dimensional quasilinear systems”, Commun. Math. Phys., 248:1 (2004), 187–206 | DOI | MR | Zbl

[3] P. A. Burovski, E. V. Ferapontov, S. P. Tsarev, “Second order quasilinear PDEs and conformal structures in projective space”, Internat. J. Math., 21:6 (2010), 799–841 | DOI | MR | Zbl

[4] M. V. Pavlov, “Klassifikatsiya integriruemykh egorovskikh gidrodinamicheskikh tsepochek”, TMF, 138:1 (2004), 55–70 | DOI | DOI | MR | Zbl

[5] E. V. Ferapontov, L. Hadjikos, K. R. Khusnutdinova, “Integrable equations of the dispersionlass Hirota type and hypersurfaces in the Lagrangian Grassmannian”, Internat. Math. Res. Notes, 2010, no. 3, 496–535 | DOI | MR | Zbl

[6] D. J. Benney, “Some properties of long non-linear waves”, Stud. Appl. Math., 52 (1973), 45–50 | DOI | Zbl

[7] J. Gibbons, “Collisionless Boltzmann equations and integrable moment equations”, Phys. D, 3:3 (1981), 503–511 ; J. Gibbons, Y. Kodama, “A method for solving the dispersionless KP hierarchy and its exact solutions. II”, Phys. Lett. A, 135:3 (1989), 167–170 | DOI | MR | DOI | MR

[8] B. A. Kupershmidt, Yu. I. Manin, “Uravneniya dlinnykh voln so svobodnoi poverkhnostyu. I. Zakony sokhraneniya i resheniya”, Funkts. analiz i ego pril., 11:3 (1977), 31–42 ; “Уравнения длинных волн со свободной поверхностью. II. Гамильтонова структура и высшие уравнения”, Функц. анализ и его прил., 12:1 (1978), 25–37 ; D. R. Lebedev, Yu. I. Manin, “Conservation laws and representation of Benney's long wave equations”, Phys. Lett. A, 74:3–4 (1979), 154–156 ; D. R. Lebedev, “Benney's long wave equations: Hamiltonian formalism”, Lett. Math. Phys., 3:6 (1979), 481–488 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[9] V. M. Teshukov, “O giperbolichnosti uravnenii dlinnykh voln”, Dokl. AN SSSR, 284:3 (1985), 555–559 ; “Характеристики, законы сохранения и симметрии кинетических уравнений движения пузырьков жидкости”, ПМТФ, 40:2 (1999), 86–100 | MR | Zbl | Zbl

[10] V. E. Zakharov, “Uravneniya Benni i kvaziklassicheskoe priblizhenie v metode obratnoi zadachi”, Funkts. analiz i ego pril., 14:2 (1980), 15–24 ; V. E. Zakharov, “On the Benney equations”, Phys. D, 3:1–2 (1981), 193–202 | DOI | MR | Zbl | DOI | Zbl