Ion-acoustic solitons in a relativistic Fermi plasma at finite temperature

The theory of ion-acoustic solitons in nonrelativistic fully degenerate plasmas and nonrelativistic and ultra-relativistic degenerate plasmas at low temperatures is known. We consider a multi-component relativistic degenerate electron-positron-ion plasma at finite temperatures. Specifically, we focus on the intermediate region where the particle's thermal energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>k</mml:mi> <mml:mi>B</mml:mi></mml:msub> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> and the rest mass energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:msup><mml:mi>c</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> do not differ significantly, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>k</mml:mi> <mml:mi>B</mml:mi></mml:msub> <mml:mi>T</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>m</mml:mi> <mml:msup><mml:mi>c</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:mrow> </mml:math> . However, the Fermi energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>k</mml:mi> <mml:mi>B</mml:mi></mml:msub> <mml:msub><mml:mi>T</mml:mi> <mml:mi>F</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> is larger than the thermal energy and the normalized chemical energy ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>/</mml:mo> <mml:msub><mml:mi>k</mml:mi> <mml:mi>B</mml:mi></mml:msub> <mml:mi>T</mml:mi></mml:mrow> </mml:math> ) is positive and finite. Two different parameter regimes with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo>≡</mml:mo> <mml:msub><mml:mi>k</mml:mi> <mml:mi>B</mml:mi></mml:msub> <mml:mi>T</mml:mi> <mml:mo>/</mml:mo> <mml:mi>m</mml:mi> <mml:msup><mml:mi>c</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , relevant for astrophysical plasmas, are defined, and the existence of small amplitude ion-acoustic solitons in these regimes are studied, including the critical cases where the known KdV (Korteweg-de Vries) theory fails. We show that while the solitons with both the positive (compressive) and negative (rarefactive) potentials coexist in the case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , only compressive solitons can exist in the other regime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . Furthermore, while the rarefactive solitons within the parameter domains of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ξ</mml:mi></mml:math> can evolve with increasing amplitude and hence increasing energy, the energy of compressive solitons reaches a steady state.