The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves

Руденко, О. В.; Hedberg, Claes

doi:10.1007/s11071-016-2721-5

Cited by 10 publications

(6 citation statements)

References 24 publications

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“…These materials exist in nature-one example is cracked or granular materials [22]. Modular and other related unusual nonlinearities exist also in composites and meta-materials, increasing the applicative interest of these studies [23], and the mathematical and physical aspects of the nonlinear dynamics in such materials will be expanded in the nearest future.…”

Section: Resultsmentioning

confidence: 99%

A new equation and exact solutions describing focal fields in media with modular nonlinearity

Руденко

Hedberg

2017

Nonlinear Dyn

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Brand-new equations which can be regarded as modifications of Khokhlov-ZabolotskayaKuznetsov or Ostrovsky-Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting because of two reasons: (1) the equations admit exact analytic solutions and (2)

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Section: Resultsmentioning

confidence: 99%

A new equation and exact solutions describing focal fields in media with modular nonlinearity

Руденко

Hedberg

2017

Nonlinear Dyn

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“…The evolution of one period of a continuous sinusoidal input wave [20] generalizing Figure 11 is shown in Figure 15. As for periodic waves described by the quadratic Burgers equation, a universal sawtooth profile forms at large distances.…”

Section: Burgers Equation With Qc Nonlinearitymentioning

confidence: 99%

Single shock and periodic sawtooth-shaped waves in media with non-analytic nonlinearities

Руденко

Hedberg

2018

Math. Model. Nat. Phenom.

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The review of new mathematical models containing non-analytic nonlinearities is given. These equations have been proposed recently, over the past few years. The models describe strongly nonlinear waves of the first type, according to the classification introduced earlier by the authors. These models are interesting because of two reasons: (i) equations admit exact analytic solutions, and (ii) solutions describe the real physical phenomena. Among these models are modular and quadratically cubic equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko type. Media with non-analytic nonlinearities exist among composites, meta-materials, inhomogeneous and multiphase systems. Some physical phenomena manifested in the propagation of waves in such media are described on the qualitative level of severity.

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“…This class of inverse problems is considered by the authors recently-the recovering of the function of the argument of a spatial variable (that determines the properties of the medium) from the data of observations of the function of the argument of the time variable (that determined by the dynamics of the reaction front). In this paper, we consider the question of the possibility of recovering the advection coefficient in the generalized Burgers equation [42] from the data on the dynamics of the reaction front.…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

et al. 2021

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The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

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The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves

Cited by 10 publications

References 24 publications

A new equation and exact solutions describing focal fields in media with modular nonlinearity

A new equation and exact solutions describing focal fields in media with modular nonlinearity

Single shock and periodic sawtooth-shaped waves in media with non-analytic nonlinearities

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

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