Variable-coefficient Kadomtsev Petviashvili Equation Research Articles

The localized excitation on the Weierstrass elliptic function periodic background for the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation

In this paper, the linear spectral problem associated with the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (gvcKP) equation with the Weierstrass function as the external potential is investigated based on the Lamé function, from which some new localized nonlinear wave solutions on the Weierstrass elliptic ℘-function periodic background are obtained by the Darboux transformation. The degenerate solutions on the ℘-function periodic background for the gvcKP equation can be derived by taking the limits of the half-periods ω 1, ω 2 of ℘(x), whose evolution and corresponding dynamics are also discussed. The findings show that nonlinear waves on the ℘-function periodic background behave as different types of nonlinear waves in different spaces, including periodic waves, vortex solitons and interaction solutions, aiding in elucidating some physical phenomena in the related fields, such as the physical ocean and nonlinear optics.

Some new lump molecules and hybrid molecular states of a (3 + 1)-dimensional generalized variable coefficient Kadomtsev–Petviashvili equation

In this paper, a (3 + 1)-dimensional generalized variable coefficient Kadomtsev–Petviashvili equation is investigated systematically, which can characterize evolution of the long water waves and small amplitude surface waves with the weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics. We investigate one lump and lump molecules obtained from one breather and breather molecules by a new degenerating breather method, respectively. In addition, the bound state of lump molecules and other localized waves is derived theoretically by velocity resonance. Considering the condition of variable coefficient, the several sets of interesting solutions having a complex structure are obtained, which include the type of parabolic, S-shaped, and periodic. The analysis method can also help us to study lump molecules existing in other integrable systems from a new perspective.

Controllable transformed waves of a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids or plasma

In this paper, we study the modulations of nonlinear transformed waves for a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids or plasma. By virtue of the phase shift analysis, the shape-changed and unchanged transformed waves are investigated, which shows the inhomogeneity can restrain the time-varying property. The deformation of waves is determined by the phase difference between two wave components. In addition, the evolutions of parabolic transformed waves are illustrated via characteristic lines analysis. The interactions are further explored, which involve the long- and short-lived collisions. In particular, we discuss the dynamics of unidirectional and reciprocating molecular waves based on the velocity resonance condition, including the shape-changed and unchanged atoms. Different from previous results, certain new types of transformed molecular waves with shape-unchanged atoms are discovered. Our results indicate that the inhomogeneity can produce novel transformed waves and further facilitate the modulation of phase transition mechanism.

VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations – variable coefficient physics-informed neural network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviate the “vanishing gradient” problem and unify linear and nonlinear coefficients. The developed method was applied to four equations including the variable coefficient Sine–Gordon (vSG), the generalized variable coefficient Kadomtsev–Petviashvili equation (gvKP), the variable coefficient Korteweg–de Vries equation (vKdV), the variable coefficient Sawada–Kotera equation (vSK). Numerical results show that VC-PINN is successful in the case of high dimensionality, various variable coefficients (polynomials, trigonometric functions, fractions, oscillation attenuation coefficients), and the coexistence of multiple variable coefficients. We also conducted an in-depth analysis of VC-PINN in a combination of theory and numerical experiments, including four aspects: the necessity of ResNet; the relationship between the convexity of variable coefficients and learning; anti-noise analysis; the unity of forward and inverse problems/relationship with standard PINN.

Hybrid-wave solutions for a (2 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics

In this paper, a (2 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics is studied. Gram-type solutions are derived via the bilinear Kadomtsev-Petviashvili hierarchy reduction method. Taking different parameter conditions in the Gram-type solutions, we construct the Y-shaped breather solutions and two types of the hybrid-wave solutions. Asymptotic forms for the aforementioned solutions are given. Based on the asymptotic forms, influences of the variable coefficients on the interactions of the breathers and solitons are studied. We obtain three types of the hybrid-wave solutions, which consist of several breathers and solitons. When those breathers and solitons interact, they form the evolving polyhedral arrangement. Changes of the entire arrangement of the breathers and solitons, and the processes of fission or fusion, are discussed and presented.

On the study of the higher-order and multiple lump/rogue waves to the (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation

In the text, rational solutions to a (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in determinant form are solved by means of the Kadomtsev-Petviashvili hierarchy reduction method and Hirota’s bilinear form. The first-order, higher-order, multi-lump waves and multi-rogue waves on the basis of the solution in Gramian are presented in graphs. Periodic and parabolic line rogue waves are obtained by choosing different dispersion coefficient functions. The second-order rogue waves are also depicted, which can be seen as the composition of two first-order rogue waves. For multiple rogue waves, taking the periodic and parabolic line rogue waves as examples, it can be observed the change tendency of amplitudes of the intersection region is different from that in branches of line rogue waves. Besides, propagation of the first-order lump wave, interaction between one-lump-soliton and one-rogue-wave, and two kinds of collisions of two lump waves, i.e., elastic and inelastic, are also shown in the plane.

Application of the polynomial function method to the variable-coefficient Kadomtsev–Petviashvili equation

In this paper, we research a (2+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. The lump, lump-soliton and lump-periodic solutions are derived based on the variable-coefficient polynomial function method. The effect of variable coefficients on the amplitude and velocity of solitons is analyzed and shown by some 3D graphs, contour plots and density graphs.

Combined Effect of Topography and Rotation on Oblique Internal Solitary Wave‐Wave Interactions

AbstractOblique interactions between internal solitary waves (ISWs) in the ocean have been frequently recorded by satellite images, nonetheless, only a limited number of studies concentrate on these phenomena, which found that both the Earth's rotation and varying topography are important factors for the underlining wave dynamics. However, the effects of rotation at different latitudes and furthermore, their combined effects with the shoaling topography are still unclear. Thus, the aim of this paper is to investigate these combined effects on oblique ISW wave‐wave interactions. Referring to observed obliquely interacting patterns, three scenarios initiated by two ISWs posing a V shape are explored using the variable‐coefficient Kadomtsev‐Petviashvili (KP) equation under the circumstance of continuous stratification and idealized shoaling topography. In all three experiments, rotation alone disintegrates the coherent waveforms and suppresses the emergence of a Mach stem (a wave generated at the interaction zone, characterized by the maximum of four‐fold initial waves). When shoaling topography is also included, which increases the nonlinearity leading to a competition with dispersion induced by rotation, and the result is a magnification of wave amplitudes and emergence of additional trailing waves. The combined effect of these two factors modulates the birth and subsequent development of the Mach stem, alters the post‐interaction patterns and determines the emergence of a nascent KdV‐like undular bore, which plausibly explains the discrepancies between previous theoretical results (when the combined effect is not considered) and the measured horizontal surface signatures for ISW wave‐wave interactions.

Wronskian solutions and Pfaffianization for a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma

In this paper, we investigate a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (GVCKP) equation in a fluid or plasma. The Nth-order Wronskian solutions for that equation are derived and proved under certain variable-coefficient constraints, where N is a positive integer. One-, two-, and three-soliton solutions in the Wronskian for that equation are given. By means of the Pfaffianization procedure, a coupled (3 + 1)-dimensional GVCKP system is constructed from that equation. Bilinear form for that coupled system is exported. Under certain variable-coefficient constraints, those Wronski-type and Gramm-type Pfaffian solutions for that coupled system are obtained and proved with the help of the Pfaffian identities.

Multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev--Petviashvili equation

Abstract Kadomtsev–Petviashvili equation is used for describing the long water wave and small amplitude surface wave with weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics. Based on the modified symbolic computation approach, the multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation are investigated. When the variable coefficient selects different functions, the dynamic properties of the derived solutions are displayed and analyzed by different three-dimensional graphics and contour graphics.

Nonlinear dynamics for different nonautonomous wave structures solutions of a (4+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Nonlinear dynamics for different nonautonomous wave structures solutions of a (4+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Study on double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients

An improved test function method for finding double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients is proposed and applied to solve the variable-coefficient Kadomtsev–Petviashvili equation, which describes waves in ferromagnetic media and matter-wave pulses in Bose–Einstein condensates.

SOLITARY AND LUMP WAVES INTERACTION IN VARIABLE-COEFFICIENT NONLINEAR EVOLUTION EQUATION BY A MODIFIED ANSÄTZ WITH VARIABLE COEFFICIENTS

<abstract> In this work, we examine variable-coefficient nonlinear evolution equations that often describe complex physical models more than constant coefficient models. A modified ansätz with variable coefficients is used for studying the solitary and lump waves interaction in these variable-coefficient nonlinear evolution equations. We discuss the variable-coefficient Kadomtsev-Petviashvili equation to achieve this goal. We present lump wave and interaction solutions between solitary and lump waves for this model. By choosing appropriate values of the variable coefficients, 3d plots and corresponding contour plots are drawn to illustrate the dynamical behaviors of the obtained solutions. </abstract>

Studies on the breather solutions for the $$\mathbf{(2+1)}$$-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Studies on the breather solutions for the $$\mathbf{(2+1)}$$-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Lumps and interaction solutions to the (4 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

In this paper, we introduce a new nonlinear evolution equation, which is ([Formula: see text])-dimensional variable-coefficient Kadomtsev–Petviashvili equation. First, according to the Hirota bilinear method, we get some exact solutions of the equation, including lump solution, lump-soliton solution, rogue-soliton solution and lump-kink solution. Then, we obtain some new exact solutions by generalizing the form of the lump solution on a further solution. Finally, based on the symbolic calculation method with Mathematica, the characteristics of the interaction solutions are shown in the graphs and we analyze the dynamic change of the solutions. Furthermore, we discuss the applications of these solutions in physics via the analysis.

Variable-coefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients

In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, a (2+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation is investigated. The multiple rogue wave solutions are obtained and their dynamics features are shown in some 3D and contour plots.

Interaction solutions of a variable-coefficient Kadomtsev–Petviashvili equation with self-consistent sources

Abstract Under investigation is a generalized variable-coefficient Kadomtsev–Petviashvili equation with self-consistent sources. Our main job is divided into four parts: (i) lump wave solution, (ii) interaction solutions between lump and solitary wave, (iii) breather wave solution and (iv) interaction solutions between lump and periodic wave. Furthermore, the interaction phenomenon of waves is shown in some 3D- and contour plots.

Nonautonomous lump solutions for a variable–coefficient Kadomtsev–Petviashvili equation

Based on the Hirota bilinear method, this paper investigates the nonautonomous lump solutions and mixed lump–stripe soliton solutions of a variable–coefficient Kadomtsev–Petviashvili equation. The displacements, trajectories and velocities of the nonautonomous lump wave are investigated by taking the variable–coefficient as the simplest power function, exponential function and sine function, respectively. In addition, the mixed lump–stripe soliton solutions are found to have periodic fission and fusion phenomena.

Multiple rogue wave solutions for a variable-coefficient Kadomtsev–Petviashvili equation

The multiple rogue wave solutions method is employed for searching the multiple soliton solutions for the new variable-coefficient Kadomtsev–Petviashvili equation, which contains first-order, second-order, third-order, and fourth-order wave 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 maximum value. The physical phenomena of these gained multiple soliton solutions are analysed and indicated in figures by selecting suitable values.

Various exact analytical solutions of a variable-coefficient Kadomtsev–Petviashvili equation

Under investigation is a generalized (3 + 1)-dimensional variable- coefficient Kadomtsev– Petviashvili equation in fluid mechanics. Various exact analytical solutions are obtained by Hirota’s bilinear method, such as lump-type, breather wave and kink-solitary wave solutions. We discuss the interaction between lump wave and solitary waves, and the interaction between lump wave and periodic wave. The physical structure and propagation characteristics of obtained solutions are shown by some 3D graphics.