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The main orientation of the current research is to look into the generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation (BKPE) for the water waves. By exerting the Cole-Hopf transform, we extract its Hirota bilinear equation. First, the weight algorithm (WA) together with the linear superposition principle(LSP) is carried out to look for the resonant multiple soliton solutions (RMSSs). Two different types of the RMSSs are obtained by introducing the parameterization of the wave numbers and frequencies. Second, the N -soliton solutions (NSSs) are also explored by using Hirota bilinear equation. On this basis, the resonance conditions of the soliton molecules on the (x, y)-, (x, z)- and (y, z)-planes are extracted and the soliton molecules are found. Finally, the ansatz function scheme, together with the symbolic computation is manipulated to look into the interaction solutions (ISs). Two different interaction solutions of the sin-cosh type and cos-cosh type are developed. A comparison between the RMSSs and the N -soliton solutions are elaborated in detail. Additionally, the dynamics of the solutions are displayed graphically to expound the physical interpretation. The proposed methods in this work can be also employed to inquire into the similar exact solutions of the other PDEs.

The solving of the derivative nonlinear Schrödinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, and also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the basis of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second-order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters is also exhibited in the simulation of the two-order rogue wave solution.

In this paper, we develop two new fourth-order integrable equations represented by nonlinear PDEs of second-order derivative in time t . The new equations model both right- and left-going waves in a like manner to the Boussinesq equation. We will employ the Painlevé analysis to formally show the complete integrability of each equation. The simplified Hirota’s method is used to derive multiple soliton solutions for this equation. We introduce a complex form of the simplified Hirota’s method to develop multiple complex soliton solutions. More exact traveling wave solutions for each equation will be derived as well.

We investigate two Boussinesq equations where the fourth-order terms come with minus and plus signs. We show that the Boussinesq equation with minus fourth-order term gives multiple soliton solutions, whereas the model with the plus fourth-order term gives multiple complex soliton solutions. We show that the two models are characterized by real and complex dispersion relations, respectively. Moreover, we derive other solitonic, singular, and periodic solutions for each model.

This study analyzes the -dimensional Boussinesq equation, a fundamental model in coastal and ocean engineering for describing the propagation of long waves in shallow water. Understanding the nonlinear wave structures of this equation is essential for predicting energy localization, wave stability, and extreme events such as rogue waves. To this end, the Hirota bilinear method is employed to derive explicit -soliton solutions, explicitly classifying them into bright and dark types according to parameter criteria. Breather solutions in different planes are constructed using the complex conjugate approach, while the long-wave limit method is applied to obtain first- and second-order lump waves, representing rationally localized structures. Furthermore, four hybrid solutions combining solitons, lumps, and breathers are developed, and their interaction dynamics (e.g. soliton–soliton and soliton–lump collisions) are systematically analyzed. The interactions are shown to be elastic, and all structures retain their identities after collision. A novel contribution of this work is the use of a bidirectional scatter plot technique to compare the behaviors of these solutions across parameter ranges, providing a unified framework for identifying conditions under which different solutions exhibit similar dynamics. The results demonstrate several practical insights: for example, lump solutions preserve their localization over time, modeling stable energy concentrations, while soliton–breather interactions capture oscillatory instabilities relevant for predicting extreme wave events. These contributions extend beyond previous studies by offering both a systematic taxonomy of nonlinear wave structures and a diagnostic tool for engineers to evaluate wave interactions under varying oceanic conditions.

In this paper, we construct the Darboux transformation of the coupled generalized Sasa-Satsuma equation using the gauge transformation between Lax pairs and obtain abundant exact solutions. This method allows obtaining many solutions by iteration, and getting the expression for the N-fold soliton solution directly. By choosing the proper parameters, some attractive solutions, including hump-type and breather-type soliton solutions, are explicitly obtained and graphically illustrated.

The present study employs the improved F-expansion and modified exp(-Z(ς))-expansion function methodologies to generate an enormous amount of novel wave solutions for the (2+1)-dimensional extended shallow water wave equation. This equation has widespread applications in many scientific and engineering domains, such as oceanography, hydraulic engineering, flood risk assessment, coastal engineering, tsunami modeling, environmental monitoring, and research on climate change. The improved F-expansion technique yields traveling wave solutions in the trigonometric and hyperbolic trigonometric formats while modified exp(-Z(ς))-expansion function generates these solutions in the rational and linear forms in addition to trigonometric and hyperbolic trigonometric forms. In this regard, a wide range of solutions, which incorporate the kink pattern, Z-pattern, singular bell pattern or singular bright, anti-kink pattern, singular anti-bell pattern or singular dark, singular periodic pattern, singular pattern, singular complexiton pattern and plane pattern solitary wave solutions are generated via these two techniques. The physical significance of the solitons and singular solitons solutions that originated using these two analytical approaches are also discussed in this work. This work also discusses the stability analysis of the model. Applying the earlier described approach, provide a number of graphical representations, such as surface, 2D, and contour graphics, that illustrate the computational and fluctuating characteristics of the produced solutions.

In this paper, a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system is investigated in fluid mechanics via the symbolic computation. With the help of the Hirota method, we derive some singular soliton, shock-wave, breather-stripe soliton and hybrid solutions. Based on the finite difference method, we get some numerical one-soliton solutions. We graphically show the singular and shock-wave solutions, and observe that the singular one-soliton solutions are explosive and unstable, but the shock-wave solutions are non-singular and stable. We observe that the breather-stripe soliton moves along the negative direction of the y axis, where y is a variable, and the amplitude and shape of the breather-stripe soliton remain invariant during the propagation. We graphically demonstrate the interaction among a rogue wave, a periodic wave and a pair of the stripe solitons: the rogue wave arises from the one stripe soliton; the rogue wave interacts with the periodic wave, the rogue wave splits into two waves and then the two waves merge into a wave; the rogue wave fuses with the other stripe soliton. We graphically present the numerical one-soliton solutions which agree with the analytic one-soliton solutions.

In this paper, we study the system of the two-mode coupled mKdV using the simplified bilinear method. We find the necessary conditions that make the solutions exists. In addition, we investigate the multiple soliton and multiple singular soliton solutions of this system. We find the necessary conditions to have N-soliton solutions. To verify the efficiency of our approach, we apply the trigonometric-function methods. The trigonometric-function methods produce 27 different solutions to this system. These solutions are the same solutions that are produced by the simplified bilinear method. Up to our knowledge, this study is new and we can apply the same idea to the other coupled systems.

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, N -soliton solutions in the Gramian are derived, where N = 1, 2, 3…. With N = 3, three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. h ( t ), l ( t ), q ( t ), n ( t ) and m ( t ), on the soliton fission and fusion are revealed: soliton velocity is related to h ( t ), l ( t ), q ( t ), n ( t ) and m ( t ), while the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h ( t ), l ( t ) and q ( t ) give the perturbed effects, and m ( t ) and n ( t ), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.