Depinning of discommensurations for tilted Frenkel–Kontorova chains

Abstract For an untilted Frenkel–Kontorova chain and any rational <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> , Aubry and Mather proved there are minimising equilibrium states that are left- and right-asymptotic to neighbouring pairs of spatially periodic minimisers of type ( p , q ). They are known as discommensurations (or kinks or fronts), advancing if the right-asymptotic equilibrium is to the right of the left-asymptotic one, retreating otherwise. Following work of Middleton, Floria &amp; Mazo and Baesens &amp; MacKay, there is a threshold tilt <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mtext>⩾</mml:mtext> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> up to which there continue to be periodic equilibria of type ( p , q ) and above which there is a globally attracting periodically sliding solution in the space of sequences of type ( p , q ). In this paper, we prove that there are values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>±</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> of tilt with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>⩽</mml:mtext> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>±</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mtext>⩽</mml:mtext> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , generically positive and less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , up to which there continue to be equilibrium advancing or retreating discommensurations, respectively, and such that for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>±</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mi>F</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> there are periodically sliding discommensurations, apart perhaps from exceptional cases with both a degenerate type ( p , q ) equilibrium and a degenerate advancing equilibrium discommensuration. We give examples, however, to show that equilibrium and periodically sliding discommensurations may co-exist, both above and below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>±</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , so the case of discommensurations is not as clean as that of periodic configurations. On the way, we prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>±</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:m