| Summary: | In this paper, we study various cosmological models involving new nonlinear forms of interaction between cold dark matter (DM) and dark energy (DE) assuming that DE is a barotropic fluid. The interactions are nonlinear either due to <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">log</mo> <mo>(</mo> <msub> <mi>ρ</mi> <mrow> <mi>d</mi> <mi>e</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>ρ</mi> <mrow> <mi>d</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> or <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">log</mo> <mo>(</mo> <msub> <mi>ρ</mi> <mrow> <mi>d</mi> <mi>m</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>ρ</mi> <mrow> <mi>d</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> parameterizations, respectively. The main purpose of this paper is to demonstrate the applicability of the forms of suggested interactions to the problem of modern cosmology known as accelerated expansion of the Universe. Using the differential age of old galaxies expressed in terms of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> data, the peak position of baryonic acoustic oscillations (known as BAO data), the SN Ia data with strong gravitational lensing data, we obtain the best fit values of the model parameters for each case. Besides, using <inline-formula> <math display="inline"> <semantics> <mrow> <mi>O</mi> <mi>m</mi> </mrow> </semantics> </math> </inline-formula> analysis and <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mn>3</mn> </msub> </semantics> </math> </inline-formula> parameter from the statefinder hierarchy analysis, we also demonstrate that the considered models are clearly different from the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics> </math> </inline-formula>CDM model. We obtain that the models predict Hubble parameter values consistent to the estimations from gravitational lensing, which probes the expansion out to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>≤</mo> <mn>1.7</mn> </mrow> </semantics> </math> </inline-formula>. We show that, with considered models, we can also explain PLANCK 2015 and PLANCK 2018 experiment results.
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