In this paper, based on the two-dimensional zero-curvature condition, we derived a new (2 + 1)-dimensional nonlinear integrable system, which was subsequently transformed into a (2 + 1)-dimensional Harry-Dym equation. We then established a connection between the corresponding matrix Lax pair and a like KP equation, and obtained 2-soliton solutions via the Darboux transformation. The soliton profiles exhibit distinct knot-like features that highlight the intricate structure of the solutions. Finally, two conservation laws are presented for new equations.

In this work, the multivariate bilinear neural network method (MBNNM) is applied to derive exact analytical solutions for nonlinear partial differential equations (NPDEs). Specifically, a (2 + 1)-dimensional spatial symmetric nonlinear dispersive wave model (SSDWM) is investigated by integrating MBNNM with established architectures (3-2-2-1, 3-2-3-1, and 3-3-2-1), while a new 3-4-2-1 architecture is developed for further investigation. By systematically selecting generalized activation functions, diverse exact analytical solutions are obtained, with their dynamic behaviors characterized via 3D/2D plots, contour plots, and density maps. To improve the computational efficiency of MBNNM in handling complex equations, a novel matrix-based solution strategy is proposed. This strategy significantly enhances computational performance by transforming the Hirota bilinear expansion into matrices for arithmetic processing.

In this paper, a multivariate bilinear residual network method is proposed to address nonlinear evolution equations. Traditional deep neural networks often encounter issues such as increased computational resource consumption, model overfitting, and vanishing gradients when dealing with complex problems. To overcome these challenges, this paper adopts the design principle of residual networks (ResNet) by introducing “shortcut connections” to directly transmit low-level features to higher layers, thereby enhancing the internal interactions within the network without increasing additional parameters or model complexity. This approach not only simplifies the model’s complexity but also improves the interactivity of the results. Furthermore, the paper elaborates on the specific steps for solving nonlinear evolution equations using the multivariate bilinear residual network method, including how to solve for weights and thresholds through symbolic computation techniques. The proposed multivariate bilinear residual network method effectively enhances training efficiency by employing “shortcut connections” to circumvent gradient vanishing issues, providing a novel approach for tackling high-dimensional complex nonlinear problems. To validate the effectiveness of this method, the paper takes the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation as an example and selects models with configurations of 3-2-2-1, 3-2-3-1, and 3-3-2-1 for investigation. By choosing different activation functions, precise analytical solutions with arbitrary activation functions are obtained, and the dynamic characteristics of these solutions are presented in three-dimensional and density plots.

Current marine-engineering and ocean-dynamics studies have been very active. On account of marine engineering, ocean dynamics, fluid mechanics, plasma physics and nonlinear optics, we hereby study a (2+1)-dimensional generalized variable-coefficient Date-Jimbo-Kashiwara-Miwa equation, for which we build up certain auto-Bäcklund transformation via a noncharacteristic movable singular manifold, solitonic solutions, analytic solutions as well as similarity reductions. As for the wave amplitude, our results depend on the variable coefficients, some of which denote the dispersion in space and space-time, separately, while some of which are caused by the geometric or physical inhomogeneities, such as the changing radius and medium density. No variable-coefficient constraints are involved in the analysis. This work may be of some theoretical use in assisting the future studies in marine engineering, ocean dynamics, fluid mechanics, plasma physics and nonlinear optics.

A variable-coefficient bilinear neural network method is proposed for deriving analytical solutions to the variable-coefficient Kadomtsev–Petviashvili equation and the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. By constructing adaptive 3-2-2-1 and 3-3-2-1 neural network models, we present abundant analytical solutions for these equations. Additionally, a variable-coefficient positive quadratic function method is introduced to obtain abundant lump-type solutions for the Sawada–Kotera equation. Dynamic properties of the derived solutions, including amplitude evolution and spatial interactions, are visualized through three-dimensional surface plots and density graphics, revealing their nonlinear wave behaviors.

Almost all the modern electronic devices have the ferromagnetic parts. For a generalized variable-coefficient Kraenkel-Manna-Merle system in a deformed ferrite, which is a generalized variable-coefficient ultrashort wave model, this paper, around a noncharacteristic movable singular manifold, symbolically computes out two magnetic auto-B & auml;cklund transformations, along with two families of the magnetic soliton solutions. Those results are in their reliance on the coefficients in that system. Electrodynamical implications of our magnetic auto-B & auml;cklund transformations and magnetic solitons come from the follow factors: the magnetic field, magnitude of the density of saturation magnetization, magnetization density, vacuum velocity of light, gyromagnetic ratio, shape of the wave during the propagation with a very short wavelength assumed, slow time variable describing the propagation over a long time or a long distance linked to the wavelength, inhomogeneity in the ferrite measuring the bond dependence of lattice defects and the corresponding exchange effects (also known as the system deformation) as well as exchange integral between the nearest-neighbor spin-spin interaction for the ferrite. With symbolic computation, the impact of inhomogeneity on the magnetic auto-B & auml;cklund transformations and on the transmission of magnetic solitons is presented. On the application side, the inhomogeneity could offer the possibility to realize the ultrafast magnetization switching in certain magnetic devices. Future experiments and observations might detect some features predicted in this paper, and relevant physical insights might be expected.

The coupled Whitham-Broer-Kaup system can be used to describe the propagation of shallow water waves in fluid dynamics. In this paper, we investigate the nonlinear conformable fractional-order Whitham-Broer-Kaup system by complex analytic method and obtain plentiful new closed-form meromorphic solutions. These derived meromorphic solutions include rational solutions, simply periodic solutions, and elliptic function solutions. At the same time, we give the real-valued characterizations of such meromorphic solutions and illustrate the dynamic behaviors of these solutions with some graphs.

This research reveals the novel types of exact wave solutions of the nonlinear Gardner–Kawahara (G–K) model in the concept of truncated M-fractional derivative. The G-K model, which is also called the extended Korteweg–de Vries (KdV) model, explains the solitary wave propagation in media, notation in plasmas, notation in shallow-water waves along surface tension and notation of magneto-acoustic waves. For our purpose, two techniques, the unified and the Sardar sub-equation techniques are applied. As a result, new types of exact wave solitons having periodic, dark–bright, periodic, kink are obtained. Some of the obtained solutions are represented through two- and three-dimensional and contour plots. The effect of the truncated M-fractional derivative (TMFD) is explained by plots. Stability of a concerned equation is checked by applying stability analysis. Moreover, the modulation instability analysis of the governing equation is also performed, which proves that the model and the obtained results are stable as well as exact.

In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein-Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters.

In this article, we will present new and more general traveling‐wave solutions for the Dodd–Bullough–Mikhailov equation based on the extended Fan sub‐equation method. The two‐dimensional affine sphere's intrinsic geometry is determined by Dodd–Bullough in the three‐dimensional unimodular affine space. The most significant accomplishment of our method is that we are able to provide all forms of the previously obtained solutions in one action, using at least four different approaches. As a result, the general elliptic equation with five parameters can be connected to other sub‐equations with three parameters that already exist, such as the Riccati equation, the first sort of elliptic equation, the auxiliary ordinary equation, and the modified Riccati equation. The extended Fan sub‐equation method is utilized to construct traveling‐wave solutions, solitary wave solutions, dark soliton solutions, and bell soliton solutions for the nonlinear Dodd–Bullough–Mikhailov equation. Mathematica is utilized to depict the dynamics of various wave structures in 3D, 2D, and contour formats, utilizing a specific set of parameters. Our graphical comparative analysis suggests that the employed method is both reliable and powerful for obtaining exact solutions of nonlinear evolution equations. The approach employed in this study may also be applicable in enhancing the understanding of numerous other complex physical phenomena.

In this work, we study a ([Formula: see text])-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. The new lump and lump-soliton solutions are obtained by the variable-coefficient polynomial function method. We used 3D graphs, contour plots and density graphs to show that the amplitude and velocity of solitons are affected by some variable coefficients. It is proved that the polynomial function method with variable coefficients is very direct and effective for solving lump-type solutions in variable coefficient integrable systems, and more new conclusions can be obtained.

This research is concerned to some modernistic solutions of new (2+1)-dimensional Korteweg–de Vries equation. The collected solutions can be executed in exposing of this model in prominent form. The obtained results including the new periodic-wave, new periodic-cross-kink wave, new three wave and other analytical wave solitons. Verification of achieved results are also done with the use of Mathematica and Maple tools. Four different techniques named as; Hirota bilinear, exp _a function, extended sinh-Gordon equation expansion and modified simplest equation techniques are employed to protect the results. The achieved results are also illustrated by 2-D, 3-D and contour plots. The gained results can also be fruitful for the development of model in future.

In this work, a variable coefficient bilinear residual network method is proposed to solve nonlinear partial differential equations with variable coefficient, including two types of neural network models: 2–2 and 3–3. Various soliton solutions of a (2+1)-dimensional coupled nonlinear equation with variable coefficients are presented based on the variable coefficient bilinear residual network method. The interaction between lump wave and solitons is discussed through a mixed function of rational and exponential functions. Finally, the interaction between lump wave and periodic wave is analyzed through a mixed function of rational and trigonometric functions. Meanwhile, some three-dimensional and density maps are used to describe the dynamic properties of the obtained results.

The Schrödinger–Hirota equation is one of the most important models of contemporary physics which is popular throughout the broad fields of fluid movement as well as in the study of thick-water crests, liquid science, refractive optical components and so on. In this paper, we utilize the Hirota bilinear technique and the unified technique to attain various soliton solutions of the governing model analytically. These approaches are robust, powerful and unique also have many applications in different fields of mathematical physics. The solutions attained from these techniques are highly valuable and useful in various fields of sciences specially in the transmissions of optical fibers, also they give different behaviors including V-shaped and periodic soliton solution behavior. Further, the approaches applied here are not applied in this model previously. Therefore, ours is a new work, which summarizes its novelty. The 3D, 2D and contour plots are included to grasp the understanding of solutions’ behavior. These findings are valuable in electronic communications such as elliptical circuits and in investigation of solitude controlling.

In this work, a more accurate analytical solution of nonlinear partial differential equation is sought by setting the generalized activation function in the model of multiple bilinear neural network method. As an example, the 3-2-2-1, 3-2-3-1, 3-3-2-1 and 3-3-3-1 models are selected to study the new (2+1) dimensional nonlinear wave equation equations. Exact analytical solutions with arbitrary activation functions are obtained by selecting different activation functions and the dynamical properties are demonstrated through three-dimensional, two-dimensional and density plots.

In this paper, a variable-coefficient symbolic computation approach is used for handling the (2+1)-dimensional variable-coefficient Hirota–Satsuma–Ito equation based on the Mathematica11 version. Abundant lump-type solutions have been obtained, including lump, lump-one soliton, lump-two solitons, lump-periodic, and lump–soliton–periodic solutions. Subsequently, the dynamic properties of these derived solutions are discussed through a large number of three-dimensional and density plots.

In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

In this work, a multivariate bilinear neural network method is proposed to seek more exact analytical solutions of nonlinear partial differential equations. As an example, the (2+1)-dimensional fractional generalized Calogero–Bogoyavlensky–Schiff–Bogoyavlensky–Konopelchenko equation is investigated via selecting the 3-2-2-1, 3-2-3-1 and 3-3-2-1 models, respectively. The exact analytical solutions with several arbitrary activation functions are derived and the dynamics properties are shown in some three-dimensional and density maps by choosing different activation functions.

The (3+1) -dimensional [ (3+1) -d] Mel’nikov equation describes an interaction between long-wave and short-wave packets in three-spatial dimensions, which is the coupling of (3+1) -d Kadomtsev–Petviashvili [KP] equation and nonlinear Schrödinger [NLS] equation. In this paper, its 𝒫𝒯 -symmetric version is introduced, and we call it (3+1) -d nonlocal Mel’nikov equation. General soliton solutions, including crossed soliton and parallel soliton, are presented in terms of KP hierarchy reduction method. Furthermore, the semi-rational solutions consisting of lumps and solitons are also constructed. These semi-rational solutions are elastic collision. However, the corresponding semi-rational solution of most nonlocal two-dimensional systems describes the inelastic collision between the lump waves and the solitons under the same conditions. Additionally, a new way to get the rational solution of Mel’nikov equation is given by reducing the semi-rational solution of the (3+1) -d nonlocal Mel’nikov equation. These novel dynamics have never been reported in (3+1) -d nonlocal systems. Moreover, it broadens our research field and inspires us to explore the mysteries of higher-dimensional nonlocal systems.