Abstract In this article, we aim to investigate some cylindrically symmetric solutions in a very well known modified theory named as $$f(R, \phi , X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> theory of gravity, where the terms R , $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> and X are clarified as Ricci Scalar, scalar potential, and kinetic term respectively. For this purpose, we consider the cylindrically symmetric space-time to discuss the cylindrical solutions in some realistic regions. We further discuss six distinct cases of exact solutions using the field equations of $$f(R, \phi , X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> modified theory of gravity. Furthermore, we set some suitable values of $$U_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> in $$f(R, \phi , X)=R+\alpha R^2 - V(\phi )+ X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> for the investigation of well-known Levi–Civita and cosmic string solutions. The Energy conditions are also investigated for all different cases and observed that null energy conditions are violated, which is the indication of the existence of cylindrical wormholes. 

/pdf/a-study-of-cylindrically-symmetric-solutions-in-f-r-phi-x-2ot2p3el.pdf

A study of cylindrically symmetric solutions in $$f(R, \phi , X)$$ theory of gravity

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Cylindrically symmetric solutions of the non-linear chiral field model (σ model)