Search results for "Dissipative dKP equation"

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Shock formation in the dispersionless Kadomtsev-Petviashvili equation

2016

The dispersionless Kadomtsev-Petviashvili (dKP) equation $(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the solutions to the transformed equation do not develop shocks. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the $(x,y)$ plane, where the solution of the dKP equation exists in a weak sense only, and a…

Shock formationFOS: Physical sciencesGeneral Physics and AstronomyKadomtsev–Petviashvili equation01 natural sciencesCritical point (mathematics)010305 fluids & plasmasDissipative dKP equation[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP]Mathematics - Analysis of PDEsMethod of characteristicsPosition (vector)[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]0103 physical sciencesFOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]0101 mathematicsSettore MAT/07 - Fisica MatematicaMathematical PhysicsMathematical physicsMathematicsCusp (singularity)Multiscales analysisdispersionless Kadomtsev-Petviashvili equation; dissipative dKP equation; multiscales analysis; shock formationPlane (geometry)Multivalued functionApplied Mathematics010102 general mathematics[ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph]Statistical and Nonlinear PhysicsMathematical Physics (math-ph)Nonlinear Sciences::Exactly Solvable and Integrable SystemsDispersionless Kadomtsev-Petviashvili equationDissipative systemAnalysis of PDEs (math.AP)
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