Semi-infinite travelling waves arising in a general reaction–diffusion Stefan model
Nabil T. Fadai
Abstract
Abstract We examine travelling wave solutions of the reaction–diffusion equation, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>∂</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mi>∂</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced close="]" open="["> <mml:mrow> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>∂</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> </mml:math> , with a Stefan-like condition at the edge of the moving front. With only a few assumptions on R ( u ) and D ( u ), a variety of new semi-infinite travelling waves arise in this reaction–diffusion Stefan model. While other reaction–diffusion models can admit semi-infinite travelling waves for a unique wavespeed, we show that semi-infinite travelling waves in the reaction–diffusion Stefan model exist over a range of wavespeeds. Furthermore, we determine the necessary conditions on R ( u ) and D ( u ) for which semi-infinite travelling waves exist for all wavespeeds. Using asymptotic analysis in various distinguished limits of the wavespeed, we obtain approximate solutions of these travelling waves, agreeing with numerical simulations with high accuracy.