Dark Lumps Research Articles

WITHDRAWN: Lump, lump with damped oscillation, double cross kinky lump with oscillation, collision of kinky and breather wave solutions to a broadened Hietarinta-like equation

One of the most significant solutions to linear and nonlinear partial differential equations is a lump soliton as it has seismic wave nature. Realizing seismic natures of such lump and oscillating lump wave, one can take initiative to mitigate its harmful energies by opposite external periodic forces. The main purpose of this study is to demonstrate various lump solutions from kinky breather-wave. In this regard, the Hirota bilinear scheme and the enhanced homoclinic test procedure are utilized to get kinky breather-wave solutions with second-order linear dispersive components that can be produced by a fourth-order nonlinear term of the Hietarinta-like equation. By making particular values of certain parameters, we derived bright- dark lump solitons from kinky-breather propagation wave solutions in a specific scenario. The cross-kinky wave in breather form, lump with damped oscillation, kink with damped oscillation, as well as lump solutions are also obtained from the Hirota bi-linear scheme constructing breather waves. A set of 3D, 2D and density graphs are used to investigate such dynamical behaviors of model numerically. These patterns are also illustrated numerous clarifications for others prototypes in the purviews of nonlinear science and engineering.

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

In this paper, we study a Generalized (2+1)-dimensional Boussinesq-type equation. Using the Hirota bilinear method, we present the N-order bright soliton solutions and dark soliton solutions. For the one-soliton solution, the bright soliton solution and the dark soliton solution share the same limit line but have different extreme values. Building on the soliton solutions, we derive higher-order bright and dark breather solutions as well as mixed solutions. The dynamic behavior is characterized using visual representations. Furthermore, through the long-wave limit method, we obtain the bright and dark lump solutions. Notably, they share the same extreme points but have different extreme values. Additionally, we derive two semi-rational solutions as lump-soliton and lump-breather. It is found that the lump moves along the peak amplitude of the soliton wave.

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Abstract The (3+1)-dimensional Geng equation is an extended version of the KdV model that describes the wave dynamics behavior of shallow water waves in complex applications. In this study, we discuss the (3+1)-dimensional Geng equation using the bilinear neural network method. By incorporating specific activation functions into the neural network model, new test functions are constructed. Using symbolic computational techniques and selecting appropriate parameters, we systematically obtain new meaningful exact solutions of some (3+1)-dimensional Geng equations, including dark lump solutions, three kinds of interaction solutions, and bright and dark soliton solutions. Furthermore, the results are visualized through diagrams of different categories, which intuitively demonstrate the evolution process and physical characteristics of the waves.

Anomalous scattering of dark lumps to the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation

The anomalous interactions of dark lumps that are completely different from the general lump collisions of the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation were constructed. Different from the normal interaction mechanisms, lumps with same velocity and depth were obtained by extending the expression of rational solution based on Hirota’s Bilinear method. Moreover, interactions between dark lumps and soliton were also investigated. In particular, rogue wave excited from the lumps and stripe soliton in the interaction duration were observed. The dynamic behavior of dark lumps was analyzed in detail, and the results obtained were illustrated graphically.

Modulational Instability and Location Controllable Lump Solutions with Mixed Interaction Phenomena for the (2+1)-Dimensional Myrzakulov-Lakshmanan-IV Equation

Under consideration is a (2+1)-dimensional spin nonisospectral Myrzakulov-Lakshmanan-IV (ML-IV) equation, which has close relation with the celebrated nonlinear Schrödinger equation. In the first place we study the modulational instability (MI) for this equation and deduce its formation mechanism for diverse localized waves from the plane wave background. Secondly, in view of the known nonisospectral Lax pair of this equation, we first successfully extend the generalized (n, N-n)-fold Darboux Transformation (DT) from (1+1)-dimensional equation to this (2+1)-dimensional equation. As an application of the resulting DT, we derive some location controllable dark and bright lump, periodic wave and mixed breather-lump solutions, it is shown that the position of these localized waves can be controlled by some special parameters, so that we can move them to the positions we want. Especially, the dynamical behaviors of these localized waves are illustrated graphically. These results may have potential value for the study of applied ferromagnetism and nano magnetism.

Breathers, interactions between breathers and lumps for a (3 + 1)-dimensional nonlinear evolution equation for the shallow-water waves

In the present paper, we investigate a -dimensional nonlinear evolution equation, which can be used to describe the shallow-water waves. Via the Kadomtsev–Petviashvili hierarchy reduction, we derive the breather, semi-rational and rational solutions in terms of Gramian. The breathers are dark breathers. Taking the long wave limit in the breather solutions, we derive the semi-rational solutions describing the interactions between the lumps and breathers, and the rational solutions which give birth to the lumps are also derived. Interactions between the lumps and breathers are elastic. Furthermore, we graphically present the interactions among the dark lumps. We find that the interactions among the three lumps are elastic.

Multiple-order line rogue wave, lump and its interaction, periodic, and cross-kink solutions for the generalized CHKP equation

The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized (2 + 1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili (CHKP) equation, which contains first-order, second-order, and third-order waves solutions. At the critical point, the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one minimum value. For the case, the lump solution will be shown the bright-dark lump structure and for another case can be present the dark lump structure-two small peaks and one deep hole. Also, the interaction of lump with periodic waves and the interaction between lump and soliton can be obtained by introducing the Hirota forms. In the meanwhile, the cross-kink wave and periodic wave solutions can be gained by the Hirota operator. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values. We alternative offer that the determining method is general, impressive, outspoken, and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering.

New computational results for a prototype of an excitable system

This present paper uses a well known computational scheme such as the modified (G'/G)-expansion method to the nonlinear predator–prey (NPP) system for forming new computational results that define a prototype of an excitable system. We construct twenty new computational solutions that define hyperbolic, trigonometric, and rational. Two-dimensional, three-dimensional, and contour shapes are depicted to demonstrate the acquired answers' more physical as well as dynamical features. Comparing our acquired responses and that obtained in previously written research articles presents the novelty of our research. The computational scheme's representation demonstrates its helpful and straightforward procedure that produces a kink-type shape, singular kink shape, bright and dark singular lump shape, multiple bright and dark lump shape, and different types of singular kink shapes. Their ability to manipulate many applications of nonlinear partial differential equations (NLPDEs) is also presented.

Lump, multi-wave, kinky breathers, interactional solutions and stability analysis for general (2 + 1)-rth dispersionless Dym equation

The Lump, multi-wave, breather, interactional solutions and stability analysis for the general r-th dispersionless Dym equation are obtained by some fruitful transformations. This approach based on an hypothesis that includes a general quadratic polynomial function with some appropriate parameters. Also for finding multi-wave, breathers and interaction phenomena we use different assumptions that includes trigonometric and exponential functions. Eventually, lump, multi-wave, bright lump, dark lump and breather wave profiles of the solutions are analyzed. These results are drawn out graphically by choosing suitable different values of parameters with detailed behavior of physical structure. At the end, we also check the stability of the governing model.

Stable and functional solutions of the Klein-Fock-Gordon equation with nonlinear physical phenomena

The present article uses a modified -expansion method and the generalized Kudryashov method on Klein-Fock-Gordon (KFG) equation and receives some stable and functional solutions. The obtained results are checked by putting backwards into the physical model and are very beneficial over various existing processes. The diverse variety of stable and functional outcomes such as kink-type shape, bright and dark lump shape, bright and dark singular kinky shape, periodic bright and dark lump shape, multiple bright and dark lump shape, the lump with rough wave shape, the rough wave shape and the kinky shape are taken. The above procedure could also be employed to get stable and functional solutions for other integral and fractional nonlinear models in physics, mathematics, and engineering.

Multiple rogue wave, lump‐periodic, lump‐soliton, and interaction betweenk‐lump andk‐stripe soliton solutions for the generalized KP equation

The multiple rogue wave solutions technique is engaged to seek the multifold soliton solutions for the generalized (2 + 1)‐dimensional Kadomtsev–Petviashvili (gKP) equation, which contains one wave, two waves, and triple waves solutions. The second‐order derivative will be perused to get the minimum or maximum amount of lump solution. For one case, the lump solution will be shown the bright‐dark lump structure, and for another case, the dark lump structure two small peaks and one deep hole can be present. Also, the interaction of lump with periodic waves and the interaction between the lump and two stripe solitons can be catched by introducing the Hirota forms. Simultaneously, the interaction betweenk‐lump andk‐stripe soliton wave solutions can be gained by the Hirota operator. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in diagrams by choosing proper amounts.

Transverse dynamics of vector solitons in defocusing nonlocal media

The transverse instability of line solitons of a multicomponent nonlocal defocusing nonlinear Schrodinger (NLS) system is utilized to construct lump and vortex-like structures in 2D nonlocal media, such as nematic liquid crystals. These line solitons are found by means of a perturbation expansion technique, which reduces the nonintegrable vector NLS model to a completely integrable scalar one, namely to a Kadomtsev–Petviashvili equation. It is shown that dark or antidark soliton stripes, as well as dark lumps, are possible depending on the strength of nonlocality: dark (antidark) solitons are formed for weaker (stronger) nonlocality, relatively to a threshold that is analytically determined in terms of the parameters of the system and the continuous-wave amplitude. Direct numerical simulations are used to show that dark lump-like- and vortex-like-structures can spontaneously be formed as a result of the transverse instability of the dark soliton stripes.

Lie symmetries and invariant solutions of $$(2+1)$$-dimensional breaking soliton equation

The present article deals with the symmetry reductions and invariant solutions of breaking soliton equation by virtue of similarity transformation method. The equation represents the collision of a Riemann wave propagating along the y-axis with a long wave along the x-axis. The infinitesimal transformations under one parameter for the governing system have been derived by exploiting the invariance property of Lie group theory. Consequently, the number of independent variables is reduced by one and the system remains invariant. A repeated application transforms the governing system into systems of ordinary differential equations. These systems degenerate well-known soliton solutions under some limiting conditions. The obtained solutions are extended with numerical simulation resulting in dark solitons, lumps, compactons, multisolitons, stationary and parabolic profiles and are shown graphically.

Localized waves and interaction solutions to an extended (3+1)-dimensional Kadomtsev–Petviashvili equation

In this paper, an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

Lump solitons and interaction phenomenon to a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

This paper studies lump solutions and interaction solutions for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation. With the help of symbolic computation and Hirota’s bilinear form, we obtain bright–dark lump solutions, lump-soliton solutions, and lump-kink solutions. Meanwhile, the dynamics of the obtained three classes of solutions are analyzed and exhibited mathematically and graphically. These results provide us with useful information to grasp the propagation processes of nonlinear waves.

Nonautonomous characteristics of lump solutions for a [formula omitted]-dimensional Korteweg–de Vries equation with variable coefficients

We study a (2+1)-dimensional Korteweg–de Vries (KdV) equation with variable coefficients. By virtue of Hirota method, we present three types of nonautonomous lump solutions including the bright, bright–dark and dark lump ones. By considering different types of dispersion coefficients, we investigate the characteristics of trajectories, velocities and displacements of nonautonomous bright lump wave, which are different from the case of its constant-coefficient counterpart. We finally demonstrate the periodic attraction and repulsion interaction between a lump wave and a soliton. Our results might provide some physical insights into the relevant fields in nonlinear science.

Dynamics of localized waves in a (3+1)-dimensional nonlinear evolution equation

In this paper, a (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equation is studied via the Hirota method. Soliton, lump, breather and rogue wave, as four types of localized waves, are derived. The obtained N-soliton solutions are dark solitons with some constrained parameters. General breathers, line breathers, two-order breathers, interaction solutions between the dark soliton and general breather or line breather are constructed by choosing suitable parameters on the soliton solution. By the long wave limit method on the soliton solution, some new lump and rogue wave solutions are obtained. In particular, dark lumps, interaction solutions between dark soliton and dark lump, two dark lumps are exhibited. In addition, three types of solutions related with rogue waves are also exhibited including line rogue wave, two-order line rogue waves, interaction solutions between dark soliton and dark lump or line rogue wave.

Dark breather waves, dark lump waves and lump wave–soliton interactions for a [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation in a fluid

Fluids are seen in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, astrophysics and biology. In this paper, we investigate a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation for the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Breather-wave, lump-wave and lump wave–soliton solutions are derived under certain conditions via the Hirota method. With h1h2−1<0, where h1 and h2 represent the coefficients of dispersion and nonlinearity, respectively, we obtain the dark breather wave and lump wave. We observe the effects of h1, h2, h4, h6 and h8 on the dark breather wave and lump wave, where h6 is the perturbed effect, h4 and h8 stand for the disturbed wave velocity effects corresponding to the y and z coordinates: h1 and h2 influence the amplitude of the dark breather wave; h1, h4 and h8 influence the distance between the adjacent valleys of the dark breather wave; h1, h4, h6 and h8 influence the location of the dark breather wave; h2, h4, h6 and h8 influence the amplitude of the dark lump wave; h1, h4 and h8 influence the width of the dark lump wave; h4, h6 and h8 influence the location of the dark lump wave. When h1h2−1>0, we present the fusion between a bright lump wave and one bright soliton as well as fission of one bright soliton. We also observe the fusion between a dark lump wave and one dark soliton as well as fission of one dark soliton with h1h2−1>0.

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

In this work, a non-isospectral and variable-coefficient Kadomtsev–Petviashvili equation is considered using Hirota’s bilinear form and a direct assumption with arbitrary functions. Analytical rational solutions in light of positive quadratic functions and lump solutions of the variable-coefficient Kadomtsev–Petviashvili equation are obtained. These lump solutions describe two types of characters by some three-dimensional graphs and contour plots, which contain bright lump wave and bright–dark lump wave. Meanwhile, periodic structure of the lump wave is also shown.

Accidental poisoning by castor bean (Ricinus communis) cake in horses

Abstract: Ricinus communis is a shrub of the family Euphorbiaceae popularly known in Brazil as “mamona” or “carrapateira”. It is an oleaginous plant whose seeds have been used mainly in biodiesel production. Plant seed oil can be extracted mechanically or using solvents, generating castor bean cake and castor bean meal as by-products, respectively. Accidental ingestion of these by-products can cause poisoning in animals and humans, characterized by digestive signs resulting from the presence of a toxalbumin called ricin. Seed toxicity varies among animal species; in horses, the lethal dose of seeds is 0.1 g/kg of body weight. The literature presents plenty of studies addressing poisoning by R. communis in different animal species and in humans; however, reports are scarce and little information is available on the pathological aspects of poisoning regarding equines. Therefore, the present study aimed to describe the epidemiological, clinical and pathological aspects observed in an outbreak of accidental poisoning with castor bean cake in horses. The equines were supplemented with palm kernel (Elaeis guianeesis) cake; however, in the purchase of a new batch, there was an error in the order, and castor bean cake was requested instead. Four horses that received castor bean cake presented clinical signs indicative of colic, which started about 21 hours after administration. Three of these horses died, with clinical evolution from 2 to 4.5 hours; the other animal was treated with intravenous fluid and antibiotic therapy and non-steroidal anti-inflammatory drugs and recovered after five days. In the necropsy of two of these horses, the main lesions were found in the small intestine, where mucosae with pronounced redness and covered with a thin layer of yellow fibrous material were observed. In the intestinal lumen, there was a large amount of bloody liquid. The stomach was full, and dark lumps similar to crushed castor bean seeds were observed in the contents. The adrenal glands of both horses presented congestion and hemorrhage in the cortex. In microscopy, the main lesion was an acute, diffuse and accentuated fibrin-necrotic enteropathy affecting the jejunum. The diagnosis of castor bean cake poisoning was based on the circumstantial evidence of by-product consumption supported by clinical and pathological aspects. The results indicate that castor bean cake commercially available as fertilizer is extremely toxic when ingested, and there is a need for appropriate detoxification or labeling informing of its toxicity.