Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

Abstract

This Memoir concerns fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow plus-or-minus normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>±</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\to \pm \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the “channels”), <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the “shelves”), and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the “exterior domain”) contains the rescaled variables. On the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the amplitude is small and the solution is highly oscillatory, while on the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> resembles that on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>but without the oscillatory correction term</italic>. Solutions of other continuous orders behave quite differently on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.