A generalized (1+2)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation: Multiple exp-function algorithm; conservation laws; similarity solutions
01 Mar 2022-Communications in Nonlinear Science and Numerical Simulation (Elsevier)-Vol. 106, pp 106072
TL;DR: A generalized BKP equation which is an augmentation of the Bogoyavlenskii–Schiff equation and Kadomtsev–Petviashvili equation is probed and can well mimic complex waves and their dealing dynamics in fluids.
About: This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2022-03-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Kadomtsev–Petviashvili equation & Conservation law.
Citations
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TL;DR: In this article , a new version of the solution in form of Grammian for the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equation is obtained with the aid of the polynomial function.
16 citations
TL;DR: 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 as discussed by the authors .
Abstract: 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.
5 citations
TL;DR: In this paper , the existence and uniqueness of solution of some nonlinear PDEs were investigated and the Ulam-Hyers-Rassias stability of solution was also investigated through an alternative theorem.
Abstract: In this study, firstly, through an alternative theorem, we study the existence and uniqueness of solution of some nonlinear PDEs and then investigate the Ulam–Hyers–Rassias stability of solution. Secondly, we apply a relatively novel analytical technique, the multiple exp function method, to obtain the multiple wave solutions of presented nonlinear equations. Finally, we propose the numerical results on tables and discuss the advantages and disadvantages of the method.
2 citations
References
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TL;DR: In this paper, the authors consider the evolution of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions, and they find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth.
Abstract: We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrodinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable.The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable.Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.
451 citations
TL;DR: In this article, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to associated bilinear equations.
Abstract: With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.
292 citations
TL;DR: In this article, a 2+1-dimensional modified integrable equation with attractors is presented. But it is not shown how to construct a 3-dimensional complex integrably complex system.
Abstract: CONTENTS Introduction Chapter I. Integrable equations with attractors ??1. Algebraic construction of differential equations with attractors ??2. Dynamical systems with attractors ??3. 1+1-dimensional integrable equations Chapter II. Breaking solitons in 2+1-dimensional integrable equations ??1. 2+1-dimensional integrable equation ??2. Basic lemma ??3. Breaking solitons and N-soliton solutions ??4. The second 2+1-dimensional integrable equation ??5. Connection with the Kadomtsev-Petviashvili equation ??6. Dynamics of the poles of meromorphic solutions ??7. Integrable two-dimensionalization of the Burgers equation and dynamics of singularities ??8. 3+1-dimensional integrable equation ??9. The third 2+1-dimensional integrable equation ??10. Application of the Painlev? method Chapter III. 2+1-dimensional modified integrable equation ??1. 2+1-dimensional modified equation ??2. Countable set of conservation laws ??3. Lax representation for the 2+1-dimensional modified equation (1.5) ??4. Lax representation for 2+1-dimensional equations (1.5) and (1.6) ??5. Lax representation with a Hermitian operator L ??6. Breaking solitons ??7. Evolution of scattering data ??8. Integrable extension of the KdV equation with the fourth-order Lax operator?L ??9. 3+1-dimensional complex integrable equation ??10. Integrable complexifications of the KdV and MKdV equations Chapter IV. Breaking solutions in continual limits of dynamical systems ??1. Continual limits of the Toda lattice and its two-dimensionalization ??2. Continual limits of the Fermi-Pasta-Ulam systems and their two-dimensionalization References
195 citations
TL;DR: A (3+1)-dimensional Hirota-Satsuma-Ito-like equation is introduced based on the (2+1), and Backlund transformation and corresponding exponential function solutions are deduced by virtue of the Hirota bilinear form.
172 citations
TL;DR: In this article, a 3 + 1 -dimensional nonlinear evolution equation is cast into Hirota bilinear form with a dependent variable transformation, which consists of six linear equations and involves nine arbitrary parameters.
Abstract: In this paper, a $$(3+1)$$
-dimensional nonlinear evolution equation is cast into Hirota bilinear form with a dependent variable transformation. A bilinear Backlund transformation is then presented, which consists of six bilinear equations and involves nine arbitrary parameters. With multiple exponential function method and symbolic computation, nonresonant-typed one-, two-, and three-wave solutions are obtained. Furthermore, two classes of lump solutions to the dimensionally reduced cases with $$y=x$$
and $$y=z$$
are both derived. Finally, some figures are given to reveal the propagation of multiple wave solutions and lump solutions.
161 citations