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The Short Pulse equation on the half lines: a Riemann--Hilbert approach
Abstract
In this work, we analyze the initial boundary value (IBV) problems for the Short Pulse (SP) equation uxt=u+1/6(u3)xx (classical solutions) on the half-lines x≥ 0 and x≤ 0 under the condition of sufficient decay as x→ +∞ or as x→ −∞. Using the Riemann--Hilbert (RH) problem approach, we show that these two IBV problems differ substantially regarding the boundary values required for problems to be well-posed. Namely, the solution of the ``right problem'' (x≥ 0) is uniquely determined by the initial data u(x,0), x≥ 0 alone, whereas for the ``left problem'' (x≤ 0), two boundary functions u(0,t) and ux(0,t) have to be prescribed, together with the initial data, for a well-posed problem.
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BibTeX
@misc{karpenko2026short,
title = {{The Short Pulse equation on the half lines: a Riemann--Hilbert approach}},
author = {Iryna Karpenko and Dmitry Shepelsky},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08105},
eprint = {2611.08105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08105},
}
RIS
TY - UNPB
TI - The Short Pulse equation on the half lines: a Riemann--Hilbert approach
AU - Iryna Karpenko
AU - Dmitry Shepelsky
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08105
UR - https://arxiv.org/abs/2611.08105
AB - In this work, we analyze the initial boundary value (IBV) problems for the Short Pulse (SP) equation u_xt=u+1/6(u^3)_xx (classical solutions) on the half-lines x≥ 0 and x≤ 0 under the condition of sufficient decay as x→ +∞ or as x→ −∞. Using the Riemann--Hilbert (RH) problem approach, we show that these two IBV problems differ substantially regarding the boundary values required for problems to be well-posed. Namely, the solution of the ``right problem'' (x≥ 0) is uniquely determined by the initial data u(x,0), x≥ 0 alone, whereas for the ``left problem'' (x≤ 0), two boundary functions u(0,t) and u_x(0,t) have to be prescribed, together with the initial data, for a well-posed problem.
ER -