Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics

Galaktionov, Victor A.; Svirshchevskii, Sergey

doi:10.1201/9781420011623

Cited by 229 publications

(379 citation statements)

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“…The questions regarding the 2mth-order approximation of odd-order evolution equations are related to difficult problems of smooth regularization of semigroups of discontinuous solutions and the construction of discontinuous extended semigroups occurring in the study of singularity-formation phenomena in PDEs; see [37,Ch. 6,7] and [38, for further references.…”

Section: Preliminary Survey: Some Known Models and Resultsmentioning

confidence: 99%

On higher-order viscosity approximations of odd-order nonlinear PDEs

Galaktionov

2007

J Eng Math

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Some aspects of vanishing viscosity (ε → 0 + ) approximations of discontinues solutions of oddorder nonlinear PDEs are discussed. The first problem concerns entropy solutions of the classic first-order conservation law (Euler's equation)which are approximated by solutions u ε (x, t) of the higher-order parabolic equationUnlike the classic case m = 1 (Burgers' equation), which is the cornerstone of modern theory of entropy solutions, direct higher-order approximations of many known entropy conditions and inequalities are not possible. By use of the concept of proper solutions from extended semigroup theory, it is shown that (0.2) and other types of approximations via 2mth-order linear or quasilinear operators correctly describe the solutions of two basic Riemann problems for (0.1) with initial datacorresponding to the shock (S − ) and rarefaction (S + ) waves, respectively. The second model is taken from nonlinear dispersion theory with the parabolic approximation(0.3) Similar evolution properties of ε-approximations of stationary shocks S ± (x) posed for (0.3) are established. Special "integrable" quasilinear odd-order PDEs are known to admit non-smooth compacton or peakontype solutions (e.g., the Rosenau-Hyman and FFCM equations), while for more general non-integrable PDEs such results are unknown. It is shown that the shock S − (x) for (0.3) is obtained as ε → 0 by an ODE approximation and also via blow-up self-similar solutions focusing as t → T − . For S + (x), the correspondDedicated to the memory of Professor S.N. Kruzhkov. 174 V. A. Galaktionoving smooth rarefaction similarity solution is indicated that explains the collapse of this non-entropy shock wave. A survey on entropy-viscosity methods developed in the last fifty years is included.

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Section: Preliminary Survey: Some Known Models and Resultsmentioning

confidence: 99%

On higher-order viscosity approximations of odd-order nonlinear PDEs

Galaktionov

2007

J Eng Math

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“…(2), but now without the initial conditions. Applying Galaktionov's method (see [4,5]), we seek the solution of Eq. (2) in the form…”

Section: Exact Solutions Of the Main Equationmentioning

confidence: 99%

“…In [3] Galaktionov's method (see [4,5]) has been applied to construct a family of exact periodic solutions of this equation. Such solutions can be useful to prove localization and elucidate the asymptotic behavior of various unbounded solutions.…”

Section: Introductionmentioning

confidence: 99%

Unbounded solutions of a nonlinear parabolic equation

Nikol’skii

2009

Comput Math Model

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The article investigates the equationWe derive a sufficient condition on the initial function of the corresponding Cauchy problem ensuring that the solution is unbounded (i.e., does not exist globally). We also construct a family of exact solutions of the form p t q t x ( ) ( ) cos + 2, where the coefficients p t ( ) and q t ( ) satisfy a first-order nonlinear dynamical system. A number of unboundedness conditions are obtained for the solutions of this family.

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“…Consider the generalized procedure of separation of variables for the nonlinear equation (5). For the construction of solutions of this equation, we use the ansatz…”

Section: Separation Of Variables For the Nonlinear Equation (5)mentioning

confidence: 99%

“…Note that we do not require that the function f x t ( , ) in ansatz (4) be represented in the form of the finite sum (3). Equation (5) was investigated in [5]. For b 1 = b 2 = 0 and a = 1, Eq.…”

Section: Introductionmentioning

confidence: 99%