Wronskian,Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

Application of the shallow water waves in environmental engineering and hydraulic engineering is seen. In this paper, a (3+1)-dimensional generalized nonlinear evolution equation (gNLEE) for the shallow water waves is investigated. The Nth-order Wronskian, Gramian and Pfaffian solutions are proved, where N is a positive integer. Soliton solutions are constructed from the Nth-order Wronskian, Gramian and Pfaffian solutions. Moreover, we analyze the second-order solitons with the influence of the coefficients in the equation and illustrate them with graphs. Through the Hirota-Riemann method, one-periodic-wave solutions are derived. Relationship between the one-periodic-wave solutions and one-soliton solutions is investigated, which shows that the one-periodic-wave solutions can approach to the one-soliton solutions under certain conditions. We reduce the (3+1)-dimensional gNLEE to a two-dimensional planar dynamic system. Based on the qualitative analysis, we give the phase portraits of the dynamic system.

     

Fully resonant soliton interactions in the Whitham–Broer–Kaup system based on the double Wronskian solutions

With the aim of exploring whether the (1+1)-dimensional coupled nonlinear evolution equations admit abundant soliton interactions, like the cases in the Kadomtsev–Petviashvili II equation, we in this paper study the double Wronskian solutions to the Whitham–Broer–Kaup (WBK) system. We give the parametric condition for two double Wronskians to generate the non-singular, non-trivial and irreducible soliton solutions. Via the asymptotic analysis of two double Wronskians, we show that the soliton solutions of the WBK system is in general linearly combined of fully resonant (M,N)- and (M?1,N+1)-soliton configurations. It turns out that the WBK system can exhibit various complex soliton structures which are different pairwise combinations of elastic, confluent and divergent interactions. From a combinatorial viewpoint, we also explain that the asymptotic solitons of a [(M,N),(M?1,N+1)]-soliton solution are identified by a pair of Grassmannian permutations.     

Spatiotemporal solitons on cnoidal wave backgrounds in three media with different distributed transverse diffraction and dispersion

The (3+1)-dimensional nonlinear Schrödinger equation with different distributed transverse diffraction and dispersion is studied based on the similarity transformation, and exact bright soliton solution on cnoidal wave backgrounds is derived. Moreover, three kinds of dynamical behaviors of these soliton solutions in three different dispersion/diffraction decreasing media with the Gaussian, hyperbolic, and Logarithmic profiles are discussed. Solitons interact with cnws and/or the change of characteristics of solitons by an addition of cnws are studied. Result of comparison with three media indicates that for the same parameters, the bright soliton in the Gaussian profile is compressed to the utmost degree. These results are potentially useful for future experiments in the optical communications, long-haul telecommunication networks, and Bose–Einstein condensations.     

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

In this paper, we consider a (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.     

General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation

General high-order rational solutions are derived for the (3+1)-dimensional Jimbo–Miwa equation based on the Hirota bilinear form. The solutions are presented in terms of Gram determinants; the elements of determinants are connected to Schur polynomials and have simple algebraic expressions. Their dynamic behaviors are researched using three-dimensional imagery and contour plots. It is revealed that different kinds of solutions appear in (x, y) plane and (y, z) plane. When one of these internal parameters in the rational solutions is sufficiently large, in (x, y) plane Lump solutions appear with obvious geometric structures, which are deconstructed by a first-order Lump such as triangle, pentagon, and nonagon, among others; in (y, z) plane rational line soliton solutions with maximum background amplitude changing over time appear. These findings might help us comprehend the nonlinear wave propagation processes in the many nonlinear physical systems.

     

Dark and gray solitons of ($$$$)-dimensional nonlocal nonlinear media with periodic response function

By using the standard symmetry reduction method, some exact analytical solutions including gray solitons and gray soliton lattice solutions are derived for the (\(2+1\))-dimensional nonlinear optical media with periodic nonlocal response. Furthermore, dark/gray soliton solutions and dark soliton lattice solutions are found by means of hyperbolic function expansion method and elliptic function expansion method for the nonlocal nonlinear system, respectively. It is found that two critical points exist for soliton solutions, and the switching dynamics of solitons may be described by the critical points.     

Geometrical patterns of time variable Kadomtsev–Petviashvili (I) equation that models dynamics of waves in thin films with high surface tension

Lump solutions are a prominent option for numerous models of nonlinear evolution. The intention of this research is to explore the variable coefficients Kadomtsev–Petviashvili equation. We auspiciously provide multiple soliton and M-lump solutions to this equation. Additionally, the presented results are also supplied with collision phenomena. Owing of its essential role, we employ appropriate parameter values to emphasis the physical characteristics of the provided results using 3D and contour charts. The outcomes of this work convey the physical characteristics of lump and lump interactions that occur in many dynamical regimes.

     

Novel evolutionary behaviors of localized wave solutions and bilinear auto-Bäcklund transformations for the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation

Water waves are common phenomena in nature, which have attracted extensive attention of researchers. In the present paper, we first deduce five kinds of bilinear auto-Bäcklund transformations of the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation starting from the specially exchange identities of the Hirota bilinear operators; then, we construct the N-soliton solutions and several new structures of the localized wave solutions which are studied by using the long wave limit method and the complex conjugate condition technique. In addition, the propagation orbit, velocity and extremum of the first-order lump solution on (x, y)-plane are studied in detail, and seven mixed solutions are summarized. Finally, the dynamical behaviors and physical properties of different localized wave solutions are illustrated and analyzed. It is hoped that the obtained results can provide a feasibility analysis for water wave dynamics.

     

One-dimensional optical solitons in cubic–quintic–septimal media with $$\varvec{\mathcal {}}$$-symmetric potentials

A (\(1+1\))-dimensional inhomogeneous cubic–quintic–septimal nonlinear Schrödinger equation with \(\mathcal {PT}\)-symmetric potentials is studied, and two families of soliton solutions are obtained. From soliton solutions, the amplitude of soliton is independent of the \(\mathcal {PT}\)-symmetric potential parameter k; however, the phase depends on the parameter k. The phase of soliton alters from negative to positive values at the location of center. Moreover, the evolutional behaviors of these solitons are discussed.     

A ship motion forecasting approach based on empirical mode decomposition method hybrid deep learning network and quantum butterfly optimization algorithm

Studies of the shallow water waves are active, possessing the applications in ocean engineering, marine environment, atmospheric science, etc. In this paper, we investigate a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Hirota method and symbolic computation help us work out (1) a bilinear form, (2) N-soliton solutions with N being a positive integer, (3) the higher-order breather solutions, (4) periodic-wave solutions and (5) hybrid solutions composed of one first-order breather and one soliton/two solitons. Moreover, we provide some nonlinear phenomena described by the associated solutions. All of the obtained results are determined via the time-dependent coefficients of that equation.

     

Soliton elastic interactions and dynamical analysis of a reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon

Under investigation in this paper is a discrete reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon, which may have some prospective applications in modern nanoribbon. First, we construct the infinitely many conservation laws and discrete N-fold Darboux transformation for this system based on its known Lax pair. Then bright–bright multi-soliton and breather solutions in terms of determinants are obtained by means of the resulting Darboux transformation. Moreover, we investigate soliton interactions through asymptotic analysis and analyze some important physical quantities such as amplitudes, wave numbers, wave widths, velocities, energies and initial phases. Finally, the dynamical evolution behaviors are discussed via numerical simulations. It is found that soliton interactions in this system are elastic, and their evolutions are stable against a small noise in a short period of time. Results obtained in this paper may have some prospective applications for understanding some physical phenomena.

     

Solutions and conservation laws of a (3+1)-dimensional Zakharov–Kuznetsov equation

In this paper, we study a nonlinear evolution partial differential equation, namely the (3+1)-dimensional Zakharov–Kuznetsov equation. Kudryashov method together with Jacobi elliptic function method is used to obtain the exact solutions of the (3+1)-dimensional Zakharov–Kuznetsov equation. Furthermore, the conservation laws of the (3+1)-dimensional Zakharov–Kuznetsov equation are obtained by using the multiplier method.     

Nonautonomous bright soliton solutions on continuous wave and cnoidal wave backgrounds in blood vessels

We discuss the dissipative nonlinear Schrödinger equation (NLSE) with a variable coefficient in blood vessels via a NLSE-based constructive method and obtain exact nonautonomous soliton solutions including bright soliton solutions on continuous wave (cw) and cnoidal wave (cnw) backgrounds. Moreover, the dynamical behaviors of these soliton solutions are studied. The impact of the cw background on the separating and interactive soliton behaviors is investigated. These behaviors of the soliton can be modulated by adjusting the amplitude of background wave. The propagation behaviors of solitons on the cnw background in different dispersion systems are also studied. These results are potentially useful for future experiments in various blood vessels.     

Lump,soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.

     

Stable localized spatial solitons in \mathcal {PT}-symmetric potentials with power-law nonlinearity

We derive analytical spatial soliton solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity in \(\mathcal {PT}\) -symmetric potentials. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulations. Moreover, some dynamical characteristics of these solutions, such as the phase switch, the power, and the transverse power-flow density, are also examined.     

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painlevé test shows that there are two solution branches, one of which has the resonance ?2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.     

Auto-Bäcklund transformations and soliton solutions on the nonzero background for a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schif equation in a fluid

In this paper, a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schif equation in a fluid is investigated. By the virtue of the truncated Painlevé expansion, a set of the auto-Bäcklund transformations of that equation is worked out. Based on the auto-Bäcklund transformations with certain non-trivial seed solutions, one-, two-, three- and N-soliton solutions on the nonzero background of that equation are derived with N as a positive integer. According to those two-soliton solutions, X- and inelastic-type soliton solutions are obtained. Via the asymptotic analysis, influence of the coefficients for the above equation is discussed and the interactions between the solitons are also studied. Then, those solitons and interactions are shown graphically.

     

Nonlocal symmetry and similarity reductions for a $$\varvec{}$$-dimensional Korteweg–de Vries equation

Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.     

The stability of two-dimensional spatial solitons in cubic–quintic–septimal nonlinear media with different diffractions and $${\varvec{\mathcal {}}}$$-symmetric potentials

A (2+1)-dimensional nonlinear Schrödinger equation in cubic–quintic–septimal nonlinear media with different diffractions and \({\mathcal {PT}}\)-symmetric potentials is studied, and (2+1)-dimensional spatial solitons are derived. The stable region of analytical spatial solitons is discussed by means of the eigenvalue method. The direct numerical simulation indicates that analytical spatial soliton solutions stably evolve within stable region in the media of focusing septimal and focusing or defocusing cubic nonlinearities with disappearing quintic nonlinearity under the 2D extended Scarf II potential. However, under the extended \({\mathcal {PT}}\)-symmetric potential with \(p=2\) and \(p=3\), analytical spatial soliton solutions stably evolve within stable region in the media of focusing quintic and septimal nonlinearities with defocusing cubic nonlinearity. In other cases, analytical spatial soliton solutions cannot sustain their original shapes, and they are distorted and broken up and finally decay into noise.     

Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method

In this paper, the Exp-function method with the aid of the symbolic computational system Maple is used to obtain the generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, (2+1)-dimensional Konopelchenko–Dubrovsky equations, the (3+1)-dimensional Jimbo–Miwa equation, the Kadomtsev–Petviashvili (KP) equation, and the (2+1)-dimensional sine-Gordon equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.