Exact solutions of generalized Burgers equation, describing travelling fronts and their interaction

Vladimirov, Vsevolod; Mączka, Cz.

doi:10.1016/s0034-4877(07)80142-x

Cited by 34 publications

(21 citation statements)

References 18 publications

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“…Indeed, it seems it has not been studied as yet. Non linear equations whose higher order part coincides with the second order linear wave operator appear in the work of Lagno et al [13,14], of Vladimirov and Kutafina [19] and of Vladimirov and Maczka [20] who are concerned about hyperbolic generalization of Burgers equation; however, no choice of the parameters therein comprised produce the nonlinear equation (31); indeed, all the nonlinear equations therein do include various terms with time derivatives.…”

Section: Discussionmentioning

confidence: 99%

“…If the material is centrosymmetric the last term in (20) vanishes, χ(x) ≡ 0, because the force P cannot generate an axial rotation.…”

Section: Navier's Equation and Kelvin's Problemmentioning

confidence: 99%

“…For numerical integration of the HKdV equation the pseudospectral method (PsM) [5,14,20] is applied. In a nutshell, the idea of the PsM is to approximate space derivatives by a certain global method -reducing thereby partial differential equation to ordinary differential equation (ODE) -and to apply a certain ODE solver for integration with respect to the time variable.…”

Section: Statement Of the Problem And Numerical Techniquementioning

confidence: 99%

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Mechanics of Microstructured Solids 2

Ganghoffer¹,

Pastrone²

2010

Lecture Notes in Applied and Computational Mechanics

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ForwordThis second volume of the series Lecture Notes in Applied and Computational Mechanics is the second part of the compendium of reviewed articles presented at the 11th EUROMECH-MECAMAT conference entitled "Mechanics of microstructured solids: cellular materials, fibre reinforced solids and soft tissues", which took place in Torino (Italy) in March 10-14, 2008, at the Museo Regional delle Scienze. This EUROMECH-MECAMAT conference was jointly organized by the Dipartimento di Matematica dell'Università di Torino, Italy and the INPL Institute (LEMTA, Nancy-Université, France). Prof. Franco Pastrone and Prof. Jean-François Ganghoffer were the co-chairmen.The conference brought together 50 scientists from 11 European countries, and was aimed at defining the current state of the art in the growing field of cellular and fibrous materials in Europe. Participants had interests in the constitutive models of micro-structured solids, non-linear wave propagation, setting up of models and identification of fibre reinforced solids, and soft tissue behaviour in a biomechanical context.The conference covered most of the mechanical and material aspects, grouped in the following four sessions:• Fibre reinforced materials; • Soft biological tissues;• Generalized continua: models and materials;• Non-linear wave propagation.The high quality talks showed a good balance between modelling and material aspects. An important part of the colloquium, with 12 presentations, was devoted to various aspects of the biomechanics of soft tissues, such as cell adhesion, constitutive models of soft tissues (brain; arteries), or models of blood flow.Beyond the scientific reports and the poster presentations, many informal exchanges were possible mainly during the coffee breaks and the lunches, according to the scope and the spirit of Euromech Colloquia and Conferences. Surely the meeting stimulated current researches and often the discussions were lively, informal and penetrating.

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Section: Discussionmentioning

confidence: 99%

“…If the material is centrosymmetric the last term in (20) vanishes, χ(x) ≡ 0, because the force P cannot generate an axial rotation.…”

Section: Navier's Equation and Kelvin's Problemmentioning

confidence: 99%

Section: Statement Of the Problem And Numerical Techniquementioning

confidence: 99%

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Mechanics of Microstructured Solids 2

Ganghoffer¹,

Pastrone²

2010

Lecture Notes in Applied and Computational Mechanics

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“…A variety of powerful methods have been used to obtain solutions to NLLEs and to investigate the physical properties of these solutions. Examples of these methods include the Darboux transformation method [9,10], the inverse scattering method [11], the Hirota bilinear method [13], the homogeneous balance method [14,15], the Lie group method [16,17], the direct method [18,19,20], and so on [21,22,23].…”

Section: Introductionmentioning

confidence: 99%

Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

Liu

Zheng

2018

Nonlinear Dyn

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Inspired by the works of Ablowitz, Mussliman and Fokas, a partial reverse space-time nonlocal Mel'nikov equation is introduced. This equation provides two dimensional analogues of the nonlocal Schrödinger-Boussinesq equation. By employing the Hirota's bilinear method, soliton, breathers and mixed solutions consisting of breathers and periodic line waves are obtained. Further, taking a long wave limit of these obtained soliton solutions, rational and semi-rational solutions of the nonlocal Mel'nikov equation are derived. The rational solutions are lumps. The semi-rational solutions are mixed solutions consisting of lumps, breathers and periodic line waves. Under proper parameter constraints, fundamental rogue waves and a semi-rational solution of the nonlocal Schrödinger-Boussinesq equation are generated from solutions of the nonlocal Mel'nikov equation.

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“…In order to better understand the nonlinear phenomena as well as further practical applications, it is important to seek their more exact traveling wave solutions. In the recent years, many powerful methods have been proposed for obtaining traveling solitary wave solutions to nonlinear evolution equations such as Hirota's bilinear method [18], the sine-cosine method [20,21], the exp-function method [16], the Jacobi elliptic function method [12], the auxiliary ordinary differential equation method [14], the direct algebraic method [17,22], and so on.…”

Section: Introductionmentioning

confidence: 99%

Exact traveling wave solutions for some nonlinear evolution equations

Lee¹,

Sakthivel²

2009

Communications in Mathematical Sciences

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Abstract. In this paper, we obtain many traveling wave solutions for some nonlinear partial differential equations. The modified tanh-coth method with the symbolic computation is implemented for constructing multiple traveling wave solutions for the two dimensional coupled Burger's, ZK-MEW and one dimensional Ostrovsky equations. The results reveal that the implemented technique is very effective and convenient for solving nonlinear partial differential equations arising in mathematical physics.

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Exact solutions of generalized Burgers equation, describing travelling fronts and their interaction

Abstract: Abstract. We present new analytical solutions to the hyperbolic generalization of Burgers equation, describing interaction of the wave fronts. To obtain them, we employ a modified version of the Hirota method.

Cited by 34 publications

References 18 publications

Mechanics of Microstructured Solids 2

Mechanics of Microstructured Solids 2

Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

Exact traveling wave solutions for some nonlinear evolution equations

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