The formation of nonlinear ion-acoustic waves is studied in a degenerate magnetoplasma accounting for quantized and trapped electrons. Relying on the reductive perturbation technique, a three-dimensional Zakharov–Kuznetsov (ZK) equation is derived, admitting a solitary wave solution with modified amplitude and width parameters. The stability of the ZK equation is also discussed using the k-expansion method. Subsequently, numerical analyses are carried out for plasma parameters of a dense stellar system involving white dwarf stars. It has been observed that the quantized magnetic field parameter η and degeneracy of electrons (determined by small temperature values T) affect the amplitude and width of the electric potential. The critical point at which the nature of the solitary structure changes from compressive to rarefaction is evaluated. Importantly, the growth rate of the instability associated with a three-dimensional ZK equation depends on the plasma parameters, and higher values of η and T tend to stabilize the solitons in quantized degenerate plasmas. The results of the present study may hold significance to comprehend the properties of wave propagation and instability growth in stellar and laboratory dense plasmas.

在解释量子化和俘获电子的简并磁等离子体中研究了非线性离子声波的形成。依靠还原微扰技术，推导出了三维 Zakharov-Kuznetsov (ZK) 方程，允许具有修正幅度和宽度参数的孤立波解。还使用 k 展开方法讨论了 ZK 方程的稳定性。随后，对包含白矮星的致密恒星系统的等离子体参数进行了数值分析。已经观察到量子化的磁场参数 η 和电子的简并性（由小温度值 T 确定）影响电势的幅度和宽度。评估孤立结构的性质从压缩变为稀疏的临界点。重要的是，与三维 ZK 方程相关的不稳定性的增长率取决于等离子体参数，并且 η 和 T 的较高值倾向于稳定量化简并等离子体中的孤子。本研究的结果可能对理解恒星和实验室致密等离子体中波传播和不稳定性增长的特性具有重要意义。