Orbital stability and instability of periodic wave solutions for ϕ4n -models

Abstract

Abstract In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general <?CDATA $\phi^{4n}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> -model for all <?CDATA $n\in\mathbb{N}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> . This family of periodic solutions are orbiting around the origin in the corresponding phase portrait and, in the standing case, are related (in a proper sense) with the aperiodic Kink solution that connect the states <?CDATA $-\tfrac{v}{2}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>−</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:math> with <?CDATA $\tfrac{v}{2}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle> <mml:mfrac> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:math> . In the traveling case, we prove the orbital instability in the whole energy space for all <?CDATA $n\in\mathbb{N}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> , while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all <?CDATA $n\in\mathbb{N}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> . Furthermore, as a by-product of our analysis, we are able to extend the main result in (de Loreno and Natali 2020 arXiv: 2006.01305 ) (given for a different family of equations) to traveling wave solutions in the whole space, for all <?CDATA $n\in\mathbb{N}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> .

Locations

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