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This paper proposes a new extended (3 + 1)‐dimensional Kadomtsev‐Petviashvili equation that portrays a unique dispersion effect about x,z$$ x,z $$. Its integrability is confirmed via the WTC‐Kruskal algorithm in Painlevé sense. N$$ N $$‐soliton, breather, and O$$ O $$‐type solitary wave are derived systematically at first. Then, the mixed solution composed of soliton and breather is obtained. In addition, the "long wave" limit is employed to construct rational and semi‐rational solution. The rational solution can be classified as rogue wave, T$$ T $$‐type solitary wave, and lump wave. The semi‐rational solution has the form a hybrid of two solitons, a hybrid of rogue wave and soliton, a hybrid of lump and soliton(s), and a hybrid of lump and breather. The results may help simulate complex waves and their interactions in fluid. [ABSTRACT FROM AUTHOR]

In this paper, a new extended (3 + 1)‐dimensional generalized Kadomtsev–Petviashvili (KP)–Boussinesq equation, with two additional terms utt$$ {u}_{tt} $$ and uxz$$ {u}_{xz} $$, is proposed and investigated. This new equation models both left and right going waves like the Boussinesq equation and describes the transmission of tsunami waves at the bottom of the ocean and nonlinear ion‐acoustic waves in the magnetized dusty plasm. We have constructed one set of breathers and first order periodic waves from the two‐soliton solution. Moreover, the four‐soliton solution consists of two sets of breathers, second order periodic waves, and a hybrid of breathers and periodic line waves. Then, we introduce two kinds of "long wave" limits to obtain the rational and semi‐rational solutions for N=2,3,4$$ N=2,3,4 $$. Under specific parametric constraints, the obtained rational and semi‐rational solutions are nonsingular. The rational solution can be classified as first order line rogue wave, single breather, second order line rogue wave, double breather, a hybrid of breather and line rogue wave, a hybrid of breather, and single soliton. The semi‐rational solution can be classified as the first and second order kink‐shaped rogue wave, a hybrid of breather and one (two) soliton(s), a hybrid of a set of breathers, and a single soliton (breather). In addition, we give Theorem 2.1 for the higher order rational solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilizability of heat equations with time‐dependent coefficients. The instability of the systems is illustrated by special solutions. The boundary feedback stabilization results are established by the backstepping method where the growth order of the kernel function depends on time. [ABSTRACT FROM AUTHOR]