Search results for "Dispersionless equation"

showing 3 items of 3 documents

The Hamilton–Jacobi Equation

2001

We already know that canonical transformations are useful for solving mechanical problems. We now want to look for a canonical transformation that transforms the 2N coordinates (q i , p i ) to 2N constant values (Q i , P i ), e.g., to the 2N initial values \((q_{i}^{0},p_{i}^{0})\) at time t = 0. Then the problem would be solved, q = q(q0, p0, t), p = p(q0, p0, t).

Dispersionless equationCombinatoricsPhysicsOmega equationCharacteristic equationCanonical transformationSummation equationCahn–Hilliard equationKadomtsev–Petviashvili equationHamilton–Jacobi equation
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Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations

2013

The formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers' equation and for dispersive regularizations as the Korteweg-de Vries equation, the Fourier coefficients indicate as …

Mathematics::Analysis of PDEsFOS: Physical sciencesKadomtsev–Petviashvili equation01 natural sciences010305 fluids & plasmasDispersionless equationMathematics - Analysis of PDEsSingularity0103 physical sciencesFOS: MathematicsMathematics - Numerical Analysis0101 mathematicsKorteweg–de Vries equationFourier seriesMathematicsMathematical physicsNonlinear Sciences - Exactly Solvable and Integrable Systems010102 general mathematicsMathematical analysisStatistical and Nonlinear PhysicsNumerical Analysis (math.NA)Condensed Matter PhysicsBurgers' equationNonlinear Sciences::Exactly Solvable and Integrable SystemsDissipative systemGravitational singularityExactly Solvable and Integrable Systems (nlin.SI)Analysis of PDEs (math.AP)Physica D
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A numerical approach to Blow-up issues for dispersive perturbations of Burgers' equation

2014

We provide a detailed numerical study of various issues pertaining to the dynamics of the Burgers equation perturbed by a weak dispersive term: blow-up in finite time versus global existence, nature of the blow-up, existence for "long" times, and the decomposition of the initial data into solitary waves plus radiation. We numerically construct solitons for fractionary Korteweg-de Vries equations.

Mathematical analysisMathematics::Analysis of PDEsStatistical and Nonlinear PhysicsNumerical Analysis (math.NA)Condensed Matter PhysicsBurgers' equationDispersionless equationNonlinear Sciences::Exactly Solvable and Integrable SystemsMathematics - Analysis of PDEsFOS: MathematicsMathematics - Numerical AnalysisFinite timeNonlinear Sciences::Pattern Formation and SolitonsMathematicsAnalysis of PDEs (math.AP)
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