AUTHOR=Barkov Ruslan , Shepelsky Dmitry 

TITLE=A Riemann–Hilbert approach to solution of the modified focusing complex short pulse equation

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=10

YEAR=2024

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2024.1466965

DOI=10.3389/fams.2024.1466965

ISSN=2297-4687

ABSTRACT=<p>We develop a Riemann–Hilbert approach to the modified focusing complex short pulse (mfcSP) equation</p><p><disp-formula id="E1a"><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:msub><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula></p><p>with zero boundary conditions (as |<italic>x</italic>| → ∞). We obtain a parametric representation of the solution of the initial value problem for the mfcSP equation in terms of the solution of the associated Riemann–Hilbert problem. This representation is then used for retrieving one-soliton solutions.</p>