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The properties and stability of nonlinear three-dimensional dust acoustic solitary waves are examined in a dusty plasma, having warm variational charged dust grains, hot ions and electrons, through the derivation of the Kadomtsev-Petviashvili equation using the reductive perturbation method. It is found that only rarefactive solitary waves can propagate in this system. To determine the stability of the waves, we used a method based on energy considerations to obtain a condition for stable solitary waves. It is found that this condition depends on dust charge variation, dust grains temperature, directional cosine of the wave vector k along the x-axis. Also, we study the dependance of the amplitude and the width on various plasma parameters. The findings of this investigation may be useful in understanding laboratory plasma phenomena and astrophysical situations.

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This paper introduces the fractal Kraenkel–Manna–Merle (KMM) system, which explains nonlinear short wave propagation with zero conductivity for saturated ferromagnetic materials in an external field. Fractal models deal with the discontinuous geometry of physical problems. The semi‐inverse technique and the new auxiliary equation method (NAEM) are used to generate a variety of solutions. A collection of exact soliton solutions specifically bright, dark, singular‐shaped, and singular‐periodic are generated using constraint conditions. The fractal parameter impact on these solutions displayed through 2D, 3D, and contour plots taking appropriate parametric values. The arbitrary functions in the solutions are chosen as such unique functions to generate some novel soliton structures. The proposed methods are more straightforward, succinct, and accurate to extract solitons of numerous evolution equations in mathematical physics.