From weak discontinuities to nondissipative shock waves

Garifullin, R. N.; Suleǐmanov, B. I.

doi:10.1134/s1063776110010164

Cited by 18 publications

(20 citation statements)

References 10 publications

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“…As it was mentioned in [28], this integral satisfies linear parts of KdV equation (7) and ODE (8). The coefficients of this system were defined in view of the condition that the pair of equations PISM (11) corresponding to simultaneous solutions of (7) and ODE (8) should possesses a fundamental solution ( , , ) with the only essential singular point = ∞ having the asymptotics…”

Section: Remarkmentioning

confidence: 99%

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“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Pavlenko¹,

Suleǐmanov²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. We consider two compatible linear evolution equations with times 1 and 2 depending on two spatial variables. These evolution equations are the analogues of the non-stationary Schrödinger equations determined by the two Hamiltonians 7 2 +1 ( 1 , 2 , 1 , 2 , 1 , 2 ) ( = 1, 2) of the Hamilton system +1 . They arise by the formal replacement of the Planck constant by the imaginary unit. We construct explicit solutions of these analogues of Schrödinger equations in terms of the solutions of the corresponding linear systems of ordinary differential equations in the isomonodromic deformations method, whose compatibility condition is the Hamiltonian system 7 2 +1 . The key role in the construction of these explicit solutions is played by the change, which was used earlier in constructing the solutions of non-stationary Schrödinger equation determined by the Hamiltonians of isomonodromic Hamiltonian Garnier system with two degrees of freedom as well as of two isomonodromic degenerations of the latter. We discuss the applicability of this change for constructing the solutions to analogues of non-stationary Schrödinger equations determined by the Hamiltonians of the entire hierarchy of isomonodromic Hamiltonian systems with two degrees of freedom being the degenerations of this Garnier system. We mention also a relation of solutions to Hamilton systems +1 with some problems of modern nonlinear mathematical physics. In particular, we show that the solutions of these Hamiltonian systems are determined explicitly by the simultaneous solutions to the Korteweg-de Vries equation + + = 0 and a non-autonomous fifth order ordinary differential equations, which are used in universal description of the influence of a small dispersion on the transformation of weak hydrodynamical discontinuities into the strong ones.

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Section: Remarkmentioning

confidence: 99%

“…In the general situation, by means of these solutions, the influence of a small dispersion on the transformation of weak hydrodynamical discontinuities into the strong one is described [28]- [30].…”

Section: Introductionmentioning

confidence: 99%

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Pavlenko¹,

Suleǐmanov²

2017

Ufimskii Matematicheskii Zhurnal

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“…(1) (in the same sense as was shown in [8][9][10][11] that the cor responding regimes are in general position for solu tions of linearized fluid dynamics equations).…”

Section: (17)mentioning

confidence: 71%

“…Such singularities are the object of ongoing research (see, e.g., [1][2][3][4][5][6][7][8][9][10][11]). Of special interest are singularities typical of solutions in general position.…”

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confidence: 99%