Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

Abstract

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L cubed"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm, resulting in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L cubed"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for DP equation, which cannot be used to control the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L cubed"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> orbital norm of the perturbation is bounded above by a function of its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> orbital norm, yielding the higher order control and the orbital stability in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared intersection upper L Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> ∩ </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2\cap L^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space.

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