Narrow Results

The $(2+1)$-dimensional extended Korteweg–de Vries (KdV) equation is predominantly utilized to elucidate the propagation of waves that exhibit both dispersive and nonlinear characteristics within the domain of nonlinear physics. This paper employs the bilinear neural network method (BNNM) to derive the exact analytical solutions of the equation. By constructing various bilinear neural network models, we obtain lump solution, breather solution and periodic interaction solution of the equation. The bilinear residual network method (BRNM) is an extension of BNNM. We apply BRNM under specific constraints to obtain breather solution of the equation, thereby offering a broader conceptual framework. Subsequently, various 3D plots, contour plots, density plots and x-curves are used to illustrate the physical properties and dynamic behaviors of these waves.

In this paper, the bilinear neural network method (BNNM) is used for the first time to investigate the (3+1)-dimensional fourth-order nonlinear equation and various bilinear neural network models are created by building diverse layers of neural networks. The precise analytical solutions of this equation include lump solution, lump and soliton interaction solution, breather solution, and interference wave solution. To better reveal the dynamic behavior and physical characteristics of these solutions, we constructed 3D plots, contour plots, and density plots. The method adopted in this paper helps study nonlinear partial differential equations in nonlinear dynamics, oceanography, and fluid mechanics. The obtained precise analytical solutions provide theoretical bases for solving high-order nonlinear equations and advancing research in nonlinear dynamics and their applications. Finally, we hope these solutions will offer new perspectives for interpreting nonlinear phenomena in high-order equations.

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra $gl(2,ℂ)$ to its commutative subalgebra $Z_2=ℂ[\mathrm{\Gamma }]/({\mathrm{\Gamma }}^2)$ and $\mathrm{\Gamma }={({\delta }_{i,j+1})}_{ij}$ to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ($q^{[1]}$, $r^{[1]}$) generated from the known solution ($q$, $r$). The new solutions ($q^{[1]}$, $r^{[1]}$) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.

To find the group invariant solution, a new shifted parity and delayed time reversal symmetries are used in order to construct Alice–Bob systems. With the help of simple assumptions and of the ${\widehat{P}}_s{\widehat{T}}_d$ symmetry, the intrinsic two-place model of the Sawada–Kotera (SK) system is elucidated. A new form of the $N$-soliton solution for the nonlocal SK equation is obtained, and dynamic properties of the $N$-soliton solutions with different values are discussed, respectively. In addition, the breather solution for the AB–SK system is also explicitly identified.

Starting with a plane wave seed, the order-$n$ breather for the (2+1)-D complex modified Korteweg-de Vries (cmKdV) equations is obtained by the use of Darboux transformation. The dynamic evolution of order-2 and order-3 breather solutions is shown in the form of pictures. Afterward, we obtain the order-$n$ degenerate breather solution by using the Taylor expansion concerning the limits ${\lambda }_{2k-1}\to {\lambda }_1(k=1,\dots ,n)$ and focus on the order-2 degenerate breather solution. We show the dynamic evolution with time and discuss the degradation process from a breather solution through getting ${\lambda }_3$ closer and closer to ${\lambda }_1$. Furthermore, the approximate trajectories of the order-2, order-3, order-4 degenerate breather solutions are depicted by explicit expressions, respectively.

Hirota equation is a modified nonlinear Schrödinger (NLS) equation, which takes into account higher order dispersion and delay correction of cubic nonlinearity. The propagation of the waves in the ocean is described, and the optical fiber can be regarded as a more accurate approximation than the NLS equation. Using the algebraic reductions from the Lie algebra $gl(n,ℂ)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Hirota systems. Considering the potential applications of two-mode nonlinear waves in nonlinear optical fibers, including its Lax pairs, we use the algebraic reductions of the Lie algebra $gl(2,ℂ)$ to its commutative subalgebra $Z_2=ℂ[\mathrm{\Gamma }]/({\mathrm{\Gamma }}^2)$. Then, we construct Darboux transformation of the strongly coupled Hirota equation, which implies the new solutions of $(q^{[1]},r^{[1]})$ generated from the known solution $(q,r)$. The new solutions $(q^{[1]},r^{[1]})$ furnish soliton solutions and breather solutions of the strongly coupled Hirota equation. Furthermore, using Taylor series expansion of the breather solutions, the rogue waves of the strongly coupled Hirota equation can be given demonstrably. It is obvious that different images can be obtained by choosing different parameters.

The $(3+1)$-dimensional generalized Korteweg-de Vries (KdV)-type model equation is investigated based on the Hirota bilinear method. Diversity of exact solutions for this equation are obtained with the help of symbolic computation. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting three-dimensional plots and contour plots. The obtained results are useful in gaining the understanding of high dimensional soliton-like structures equation related to mathematical physics branches, natural sciences and engineering areas.

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, $N$-soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model.

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the ($3+1$)-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields.

Bifurcations of a general class of traveling wave solutions are analyzed. In particular, the existence of solitary wave, kink and anti-kink wave solutions, and uncountably infinite periodic wave solutions and breather solutions of a general class of traveling wave equations is proved. Also, the existence of breaking wave solution is discussed in detail. Under different parametric conditions, several sufficient conditions for the existence of these solutions are derived. Sufficient simulation results are provided to visualize the theoretical results.

In this paper, we consider a model which is a generalization of the nonlinear Schrödinger equation where the dispersive term was substituted by a nonlocal integral term with given kernel. The study on this model derives a planar dynamical system with two singular straight lines. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical system, we obtain all possible explicit exact parametric representations of solutions (including kink wave solutions, unbounded wave solutions, compactons, etc.) under different parameter conditions. The existence of bounded solutions of the planar dynamical system implies that there exist infinitely many breather solutions of this generalized nonlinear Schrödinger system.