A global theory of internal solitary waves in two-fluid systems

Abstract

The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is assumed that a fluid of constant density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\rho _ + }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lies above a fluid of constant density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript minus Baseline greater-than rho Subscript plus Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>−</mml:mo> </mml:msub> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\rho _ - } &gt; {\rho _ + } &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis lamda comma w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\lambda ,w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where the constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda equals g slash c squared"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda = g/{c^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> being the wave speed, and where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis x comma eta right-parenthesis plus eta"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>η</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">w(x,\eta ) + \eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the height at a horizontal position <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the streamline which has height <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta"> <mml:semantics> <mml:mi>η</mml:mi> <mml:annotation encoding="application/x-tex">\eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals plus-or-minus normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x = \pm \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis lamda comma w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\lambda ,w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an unbounded connected set in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R times left-parenthesis upper H 0 Superscript 1 Baseline intersection upper C Superscript 0 comma 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>×</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo>∩</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {R}} \times (H_0^1 \cap {C^{0,1}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.