Asymptotic soliton-form solutions of equations with small dispersion

Maslov, Victor Pavlovich; Omel’yanov, Georgii A.

doi:10.1070/rm1981v036n03abeh004248

Cited by 105 publications

(80 citation statements)

References 44 publications

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“…These soliton solutions are similar to the delta solitons proposed and analyzed by Sarrico [35] which were inspired by the seminal works of Maslov et al [36,37].…”

Section: Background On the Burgers Equationmentioning

confidence: 65%

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

Davidson¹

2010

Physica E: Low-dimensional Systems and Nanostructures

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Quark-gluon plasmas formed in heavy ion collisions at high energies are well described by ideal classical fluid equations with nearly zero viscosity. It is believed that a similar fluid permeated the entire universe at about three microseconds after the big bang. The estimated Reynolds number for this quark-gluon plasma at 3 microseconds is approximately 10^19. The possibility that quantum mechanics may be an emergent property of a turbulent proto-fluid is tentatively explored. A simple relativistic fluid equation which is consistent with general relativity and is based on a cosmic dust model is studied. A proper time transformation transforms it into an inviscid Burgers equation. This is analyzed numerically using a spectral method. Soliton-like solutions are demonstrated for this system, and these interact with the known ergodic behavior of the fluid to yield a stochastic and chaotic system which is time reversible. Various similarities to quantum mechanics are explored.

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“…These soliton solutions are similar to the delta solitons proposed and analyzed by Sarrico [35] which were inspired by the seminal works of Maslov et al [36,37].…”

Section: Background On the Burgers Equationmentioning

confidence: 65%

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

Davidson¹

2010

Physica E: Low-dimensional Systems and Nanostructures

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“…Following by V P Maslov [25] we will use the words "asymptotic solution with respect to mod (O (ε α ))", namely, a function is said to be an asymptotic solution of mod (O (ε a )) of the equation (1) if after its substitution into this equation the latter is satisfied up to the terms of the order O (ε a ).…”

Section: The Main Resultsmentioning

confidence: 99%

“…The solution of the equation for the main term is the function 0 ρ(s) which is defined by (25). The equations for the corrections is the Lame equations with external force.…”

Section: Asymptotic Behaviour Of First Inner Expansionmentioning

confidence: 99%

Hard Loss of Stability in Painlevé-2 Equation

Kiselev¹

2001

JNMP

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A special asymptotic solution of the Painlevé-2 equation with small parameter is studied. This solution has a critical point t * corresponding to a bifurcation phenomenon. When t < t * the constructed solution varies slowly and when t > t * the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution, using a sequence of scaling and matching procedures.

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“…where χ(τ, t) is a regularization of the Heaviside function, Z 1 (τ, t) is a special function from the Schwartz space, and c 1 is arbitrary constant (see [16,17]). If it is violated and, for example,…”

Section: Lemmamentioning

confidence: 99%

Collision of solitons for a nonhomogeneous version of the KdV equation

Omel’yanov¹

2018

qpr

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We consider KdV-type equations with C 1 nonhomogeneous nonlinearities and small dispersion ε. The first result consists of the conclusion that, in the leading term with respect to ε, the solitary waves in this model interact like KdV solitons. Next, it turned out that there exists a very interesting scenario of instability in which the short-wave soliton remains stable whereas a small long-wave part, generated by perturbations of original equation, turns to be unstable, growing and destroying the leading term. At the same time, such perturbation can eliminate the collision of solitons.

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Asymptotic soliton-form solutions of equations with small dispersion

Cited by 105 publications

References 44 publications

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

The quark-gluon plasma and the stochastic interpretation of quantum mechanics

Hard Loss of Stability in Painlevé-2 Equation

Collision of solitons for a nonhomogeneous version of the KdV equation

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