A DISTRIBUTIONAL PRODUCT APPROACH TO δ-SHOCK WAVE SOLUTIONS FOR A GENERALIZED PRESSURELESS GAS DYNAMICS SYSTEM

2014 ◽  
Vol 25 (01) ◽  
pp. 1450007 ◽  
Author(s):  
C. O. R. SARRICO
Keyword(s):  
Shock Wave ◽  
Gas Dynamics ◽  
General Framework ◽  
Solution Concept ◽  
Singular Solutions ◽  
Wave Solutions ◽  
Shock Wave Solution ◽  
Consistent Extension ◽  
Distributional Product ◽  
Product Approach

In the setting of a product of distributions which is not defined by approximation processes, we are able to consider a Riemann problem for the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0, with the unknown states u, v in convenient spaces of distributions and ϕ, ψ : ℝ → ℝ continuous. A consistent extension of the classical solution concept will show the possible arising of a δ-shock wave solution. This procedure affords a simpler and more general framework to construct singular solutions and can surely be applied to other equations or systems.

Shock wave solutions of the variants of the Kadomtsev–Petviashvili equation

2011 ◽  
Vol 89 (9) ◽  
pp. 979-984 ◽  
Author(s):  
Houria Triki ◽  
B.J.M. Sturdevant ◽  
T. Hayat ◽  
O.M. Aldossary ◽  
A. Biswas
Keyword(s):  
Shock Wave ◽  
Wave Solution ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Shock Wave Solution

This study obtained the shock wave or kink solutions of the variants of the Kadomtsev–Petviashvili equation with generalized evolution. There are three types of variants of this equation that were considered. The relation between the parameters and the constraint conditions will naturally fall out as a consequence of the derivation of the shock wave solution.

Distributions as initial values in a triangular hyperbolic system of conservation laws

2019 ◽  
Vol 150 (6) ◽  
pp. 2757-2775 ◽  
Author(s):  
C. O. R. Sarrico ◽  
A. Paiva
Keyword(s):  
Cauchy Problem ◽  
Shock Waves ◽  
Conservation Laws ◽  
Classical Solution ◽  
Hyperbolic System ◽  
Initial Conditions ◽  
Solution Concept ◽  
Space Of Distributions ◽  
Consistent Extension ◽  
Distributional Product

AbstractThe present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

New δ-shock waves in the p-system: a distributional product approach

2019 ◽  
Vol 25 (3) ◽  
pp. 619-629 ◽  
Author(s):  
Adelino Paiva
Keyword(s):  
Weak Solution ◽  
Entire Function ◽  
Shock Waves ◽  
Riemann Problem ◽  
Solution Concept ◽  
P System ◽  
One Dimensional ◽  
System A ◽  
Distributional Product ◽  
Product Approach

This article studies a Riemann problem for the so-called “[Formula: see text]-system”[Formula: see text], [Formula: see text], which rules one-dimensional isentropic thermoelastic media. Such study is made using a product of distributions that allows us to extend both the classical solution concept and a weak solution concept. By considering [Formula: see text] as an entire function that takes real values on the real axis, this product also extends for certain distributions [Formula: see text] the meaning of [Formula: see text]. Under certain conditions, this Riemann problem has solutions that are [Formula: see text]-shock waves. Furthermore, those [Formula: see text]-shock waves satisfy the so-called generalized Rankine–Hugoniot conditions.

scholarly journals New Distributional Global Solutions for the Hunter-Saxton Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
C. O. R. Sarrico
Keyword(s):  
Travelling Waves ◽  
Solution Concept ◽  
Global Solutions ◽  
Travelling Wave ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Discontinuous Solutions ◽  
Consistent Extension ◽  
Distributional Product ◽  
Necessary And Sufficient

In the setting of a distributional product, we consider a Riemann problem for the Hunter-Saxton equation[ut+((1/2)u2)x]x=(1/2)ux2in a convenient space of discontinuous functions. With the help of a consistent extension of the classical solution concept, two classes of discontinuous solutions are obtained: one class of conservative solutions and another of dispersive solutions. A necessary and sufficient condition for the propagation of a distributional profile as a travelling wave is also presented, which allows identifying an interesting set of explicit distributional travelling waves. In the paper, we will show some results we have obtained by applying this framework to other equations and systems.

scholarly journals Nematic Dispersive Shock Waves from Nonlocal to Local

2021 ◽  
Vol 11 (11) ◽  
pp. 4736
Author(s):  
Saleh Baqer ◽  
Dimitrios J. Frantzeskakis ◽  
Theodoros P. Horikis ◽  
Côme Houdeville ◽  
Timothy R. Marchant ◽  
...  
Keyword(s):  
Shock Wave ◽  
Liquid Crystals ◽  
Shock Waves ◽  
Numerical Solutions ◽  
Wave Structure ◽  
Beam Power ◽  
Input Beam ◽  
Shock Wave Structure ◽  
Wave Solutions ◽  
Modulation Theory

The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonance is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations, and excellent agreement is found.

Optical solitons and stability analysis for the generalized fourth-order nonlinear Schrödinger equation

2019 ◽  
Vol 33 (27) ◽  
pp. 1950333
Author(s):  
Xiao-Song Tang ◽  
Biao Li
Keyword(s):  
Nonlinear Dynamics ◽  
Fourth Order ◽  
Continuous Wave ◽  
Modulational Instability ◽  
Optical Solitons ◽  
Singular Solutions ◽  
Nls Equation ◽  
Ansatz Method ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

We consider a generalized fourth-order nonlinear Schrödinger (NLS) equation. Based on the ansatz method, its bright, dark single-soliton is constructed under some constraint conditions. Furthermore, combining the Riccati equation extension approach, we also derive some exact singular solutions. With several parameters to play with, we display the dynamic behaviors of bright, dark single-soliton. Finally, the condition for the modulational instability (MI) of continuous wave solutions for the equation is generated. It is hoped that our results can help enrich the nonlinear dynamics of the NLS equations.

scholarly journals A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yi-Hu Feng ◽  
Lei Hou
Keyword(s):  
Shock Wave ◽  
Time Delay ◽  
Wave Solution ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Expansion Method ◽  
Initial Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Shock Wave Solution

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

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