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Uspekhi Mat. Nauk, 1981, Volume 36, Issue 3(219), Pages 63–126 (Mi umn2926)  

This article is cited in 48 scientific papers (total in 48 papers)

Asymptotic soliton-form solutions of equations with small dispersion

V. P. Maslov, G. A. Omel'yanov


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English version:
Russian Mathematical Surveys, 1981, 36:3, 73–149

Bibliographic databases:

UDC: 517.9
MSC: 37K40, 37K10, 35Q53, 35Q51
Received: 30.09.1980

Citation: V. P. Maslov, G. A. Omel'yanov, “Asymptotic soliton-form solutions of equations with small dispersion”, Uspekhi Mat. Nauk, 36:3(219) (1981), 63–126; Russian Math. Surveys, 36:3 (1981), 73–149

Citation in format AMSBIB
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\pages 63--126
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\jour Russian Math. Surveys
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\vol 36
\issue 3
\pages 73--149
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. Yu. Subochev, V. A. Tsupin, “Asymptotic soliton-like solutions of the nonlinear Schrd̈inger equation with variable coefficients”, Theoret. and Math. Phys., 56:1 (1983), 661–668  mathnet  crossref  mathscinet  isi
    2. V. P. Maslov, “Non-standard characteristics in asymptotic problems”, Russian Math. Surveys, 38:6 (1983), 1–42  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. V. P. Maslov, V. M. Chetverikov, “Asymptotic solutions of the Landau–Lifshitz equation and quasisteady motion of bubbles in magnetic films”, Theoret. and Math. Phys., 60:3 (1984), 931–944  mathnet  crossref  mathscinet  isi
    4. V. P. Maslov, “Coherent structures, resonances, and asymptotic non-uniqueness for Navier–Stokes equations with large Reynolds numbers”, Russian Math. Surveys, 41:6 (1986), 23–42  mathnet  crossref  mathscinet  adsnasa  isi
    5. S. A. Lomov, A. G. Eliseev, “Asymptotic integration of singularly perturbed problems”, Russian Math. Surveys, 43:3 (1988), 1–63  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. L. A. Kalyakin, “Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems”, Russian Math. Surveys, 44:1 (1989), 3–42  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. A. M. Il'in, “On the asymptotics of the solution of a problem with a small parameter”, Math. USSR-Izv., 34:2 (1990), 261–279  mathnet  crossref  mathscinet  zmath
    8. Yu. V. Egorov, “A contribution to the theory of generalized functions”, Russian Math. Surveys, 45:5 (1990), 1–49  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. H. A. Biagioni, M. Oberguggenberger, “Generalized Solutions to the Korteweg–de Vries and the Regularized Long-Wave Equations”, SIAM J Math Anal, 23:4 (1992), 923  crossref  mathscinet  zmath  isi
    10. L. A. Kalyakin, “Perturbation of the Korteweg–de Vries soliton”, Theoret. and Math. Phys., 92:1 (1992), 736–747  mathnet  crossref  mathscinet  zmath  isi
    11. O. M. Kiselev, “Kink asymptotics of the perturbed sine-Gordon equation”, Theoret. and Math. Phys., 93:1 (1992), 1106–1111  mathnet  crossref  mathscinet  zmath  isi
    12. L. A. Kalyakin, “On the problem of first correction in soliton perturbation theory”, Sb. Math., 186:7 (1995), 977–1002  mathnet  crossref  mathscinet  zmath  isi
    13. V. G. Danilov, G. A. Omel'yanov, E. V. Radkevich, “Justification of asymptotics of solutions of the phase-field equations and a modified Stefan problem”, Sb. Math., 186:12 (1995), 1753–1771  mathnet  crossref  mathscinet  zmath  isi
    14. V. M. Shelkovich, “Associative and commutative distribution algebra with multipliers, and generalized solutions of nonlinear equations”, Math. Notes, 57:5 (1995), 536–549  mathnet  crossref  mathscinet  zmath  isi  elib
    15. L. A. Kalyakin, “Asymptotics of the first correction in the perturbation of the $N$-soliton solution to the KdV equation”, Math. Notes, 58:2 (1995), 814–823  mathnet  crossref  mathscinet  zmath  isi
    16. R. R. Gadyl'shin, O. M. Kiselev, “On nonsolution structure of scattering data under perturbation of two-dimensional soliton for Davey–Stewartson equation II”, Theoret. and Math. Phys., 106:2 (1996), 167–173  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. N. M. Bessonov, S. A. Vakulenko, “Connected kink states in nonlinear inhomogeneous media”, Theoret. and Math. Phys., 107:1 (1996), 511–522  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. L. A. Kalyakin, V. A. Lazarev, “Perturbation of the two-soliton solution of the KdV equation”, Theoret. and Math. Phys., 112:1 (1997), 866–874  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    19. V. G. Danilov, V. P. Maslov, V. M. Shelkovich, “Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems”, Theoret. and Math. Phys., 114:1 (1998), 1–42  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    20. V.G. Danilov, V.M. Shelkovich, “Generalized solutions of nonlinear differential equation and the Maslov algebras of distributions”, Integral Transforms and Special Functions, 6:1-4 (1998), 171  crossref  elib
    21. M. Nedeljkov, “Infinitely narrow soliton solutions to scalar conservation laws in Colombeau sense”, Integral Transforms and Special Functions, 6:1-4 (1998), 257  crossref
    22. G. A. Omel'yanov, V. V. Trushkov, “A geometric correction in the problem on the motion of a free boundary”, Math. Notes, 63:1 (1998), 137–139  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. G. A. Omel'yanov, V. V. Trushkov, “Dynamics of a free boundary in a binary medium with variable thermal conductivity”, Math. Notes, 66:2 (1999), 181–189  mathnet  crossref  crossref  mathscinet  zmath  isi
    24. A. I. Shafarevich, “The Navier–Stokes equations: Asymptotic solutions describing tangential discontinuities”, Math. Notes, 67:6 (2000), 792–801  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. Shafarevich, AI, “Localized asymptotic solutions of the Navier–Stokes equations and topological invariants of vector fields. Prandtl-Maslov equations on Reeb graphs and Fomenko invariants”, Russian Journal of Mathematical Physics, 7:4 (2000), 426  mathscinet  zmath  isi  elib
    26. S. P. Kshevetskii, “Numerical simulation of nonlinear internal gravity waves”, Comput. Math. Math. Phys., 41:12 (2001), 1777–1791  mathnet  mathscinet  zmath  elib
    27. A. YU. KHRENNIKOV, V. M. SHELKOVICH, O. G. SMOLYANOV, “LOCALLY CONVEX SPACES OF VECTOR-VALUED DISTRIBUTIONS WITH MULTIPLICATIVE STRUCTURES”, Infin. Dimens. Anal. Quantum. Probab. Relat. Top, 05:04 (2002), 483  crossref
    28. K. A. Volosov, “A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations”, Math. Notes, 71:3 (2002), 339–354  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    29. Igor V Andrianov, Jan Awrejcewicz, Rem G Barantsev, “Asymptotic approaches in mechanics: New parameters and procedures”, Appl Mech Rev, 56:1 (2003), 87  crossref  elib
    30. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230  mathnet  crossref  mathscinet  zmath  elib
    31. E. V. Radkevich, “Well-posedness of mathematical models of continuum mechanics and thermodynamics”, Journal of Mathematical Sciences (New York), 125:3 (2005), 259  crossref  mathscinet  zmath
    32. V.G. Danilov, V.M. Shelkovich, “Dynamics of propagation and interaction of <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>δ</mml:mi></mml:math>-shock waves in conservation law systems”, Journal of Differential Equations, 211:2 (2005), 333  crossref
    33. E. L. Aero, S. A. Vakulenko, “Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice”, Theoret. and Math. Phys., 143:3 (2005), 782–791  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    34. Maslov, VP, “Rapidly oscillating asymptotic solutions of the Navier–Stokes equations, coherent structures, Fomenko invariants, Kolmogorov spectrum, and flicker noise”, Russian Journal of Mathematical Physics, 13:4 (2006), 414  crossref  mathscinet  zmath  adsnasa  isi  elib
    35. Kulagin, DA, “Interaction of kinks for semilinear wave equations with a small parameter”, Nonlinear Analysis-Theory Methods & Applications, 65:2 (2006), 347  crossref  mathscinet  zmath  isi  elib
    36. Eron Aero, Alexander Fradkov, Boris Andrievsky, Sergey Vakulenko, “Dynamics and control of oscillations in a complex crystalline lattice”, Physics Letters A, 353:1 (2006), 24  crossref  elib
    37. C.O.R. Sarrico, “New solutions for the one-dimensional nonconservative inviscid Burgers equation”, Journal of Mathematical Analysis and Applications, 317:2 (2006), 496  crossref
    38. S. A. Kordyukova, “Korteweg–de Vries hierarchy as an asymptotic limit of the Boussinesq system”, Theoret. and Math. Phys., 154:2 (2008), 250–259  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    39. Garcia M.G., Omel'yanov G.A., “Kink-Antikink Interaction for Semilinear Wave Equations With a Small Parameter”, Electronic Journal of Differential Equations, 2009, 45  isi
    40. Mark Davidson, “The quark-gluon plasma and the stochastic interpretation of quantum mechanics”, Physica E: Low-dimensional Systems and Nanostructures, 42:3 (2010), 317  crossref
    41. Omel'yanov G.A., Segundo-Caballero I., “Asymptotic and Numerical Description of the Kink/Antikink Interaction”, Electronic Journal of Differential Equations, 2010, 150  isi
    42. Martin G. Garcia Alvarado, Georgii A. Omel’yanov, “Interaction of solitary waves for the generalized KdV equation”, Communications in Nonlinear Science and Numerical Simulation, 2011  crossref
    43. V. P. Maslov, A. I. Shafarevich, “Asymptotic solutions of Navier–Stokes equations and topological invariants of vector fields and Liouville foliations”, Theoret. and Math. Phys., 180:2 (2014), 967–982  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    44. M. Colombeau, “Irregular shock waves formation as continuation of analytic solutions”, Applicable Analysis, 2014, 1  crossref
    45. G.A.. Omel'yanov, “Soliton-type asymptotics for non-integrable equations: a survey”, Math. Meth. Appl. Sci, 2014, n/a  crossref
    46. C. O. R. Sarrico, “The Brio system with initial conditions involving Dirac masses: A result afforded by a distributional product”, Chin. Ann. Math. Ser. B, 35:6 (2014), 941  crossref
    47. Sarrico C.O.R. Paiva A., “Products of Distributions and Collision of a Delta-Wave With a Delta `-Wave in a Turbulent Model”, 22, no. 3, 2015, 381–394  crossref  isi
    48. V. P. Maslov, A. I. Shafarevich, “Fomenko invariants in the asymptotic theory of the Navier–Stokes equations”, J. Math. Sci., 225:4 (2017), 666–680  mathnet  crossref  mathscinet  elib
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