Volume 5, Issue 3
Nonlinear Superposition of Delta Waves in Multi-dimensional Space

Fang Daoyuan

DOI:

J. Part. Diff. Eq.,5(1992),pp.72-84

Published online: 1992-05

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  • Abstract

We study the behavior of the solution u^ε to the semilinear wave equation with initial data aξ + u_i(i = 1, 2) in multidimensional space, where u_i is a classical function and aξ is smooth and converges to a distribution a_i as ε → 0. In some circumstances one can prove the convergence of u^ε, and our results express a striking superposition principle. The singular part of the solution propagates linearly. The classical part shows the nonlinear effects. And, the limit of the nonlinear solution u^ε, delta wave, as the data become more singular is the sum of the two parts.

  • Keywords

Semilinear wave equation delta wave superposition principle

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@Article{JPDE-5-72, author = {Fang Daoyuan}, title = {Nonlinear Superposition of Delta Waves in Multi-dimensional Space}, journal = {Journal of Partial Differential Equations}, year = {1992}, volume = {5}, number = {3}, pages = {72--84}, abstract = { We study the behavior of the solution u^ε to the semilinear wave equation with initial data aξ + u_i(i = 1, 2) in multidimensional space, where u_i is a classical function and aξ is smooth and converges to a distribution a_i as ε → 0. In some circumstances one can prove the convergence of u^ε, and our results express a striking superposition principle. The singular part of the solution propagates linearly. The classical part shows the nonlinear effects. And, the limit of the nonlinear solution u^ε, delta wave, as the data become more singular is the sum of the two parts.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5746.html} }
TY - JOUR T1 - Nonlinear Superposition of Delta Waves in Multi-dimensional Space AU - Fang Daoyuan JO - Journal of Partial Differential Equations VL - 3 SP - 72 EP - 84 PY - 1992 DA - 1992/05 SN - 5 DO - http://dor.org/ UR - https://global-sci.org/intro/jpde/5746.html KW - Semilinear wave equation KW - delta wave KW - superposition principle AB - We study the behavior of the solution u^ε to the semilinear wave equation with initial data aξ + u_i(i = 1, 2) in multidimensional space, where u_i is a classical function and aξ is smooth and converges to a distribution a_i as ε → 0. In some circumstances one can prove the convergence of u^ε, and our results express a striking superposition principle. The singular part of the solution propagates linearly. The classical part shows the nonlinear effects. And, the limit of the nonlinear solution u^ε, delta wave, as the data become more singular is the sum of the two parts.
Fang Daoyuan. (1970). Nonlinear Superposition of Delta Waves in Multi-dimensional Space. Journal of Partial Differential Equations. 5 (3). 72-84. doi:
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