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We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations. 相似文献
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It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well. 相似文献
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The investigation of the exact traveling wave solutions to the nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. To understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties. The Wronskian technique is a powerful tool to construct multi-soliton solutions for many nonlinear evolution equations possessing Hirota bilinear forms. In the process of utilizing the Wronskian technique,the main difficulty lies in the construction of a system of linear differential conditions, which is not unique. In this paper,we give a universal method to construct a system of linear differential conditions. 相似文献
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Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers' equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions. 相似文献
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In this paper, a two-step ansatz is proposed, which leads to some new solutions to two coupled nonlinear physical equations. 相似文献
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V.A. Vladimirov 《Reports on Mathematical Physics》2004,54(2):261-271
A direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions. 相似文献
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On the investigation of nonlinear waves of the Hirota and the Maxwell–Bloch equation in nonlinear optics
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In this paper, considering the Hirota and Maxwell-Bloch (H-MB) equations which is governed by femtosecond pulse propagation through two-level doped fibre system, we construct the Darboux transformation of this system through linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system. 相似文献
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In this Letter, to further understand the role of nonlinear dispersion in coupled nonlinear wave systems in both real and complex fields, we study the coupled Klein–Gordon equations with nonlinear dispersion in real field (called CKG(m,n,k) equation) and (2+1)-dimensional generalization of coupled nonlinear Schrödinger equation with nonlinear dispersion in complex field (called GCNLS(m,n,k) equation) via some transformations. As a consequence, some types of solutions are obtained, which contain compactons, solitary pattern solutions, envelope compacton solutions, envelope solitary pattern solutions, solitary wave solutions and rational solutions. 相似文献
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An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein-Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions. 相似文献
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Exact travelling wave solutions to some nonlinear equations of fifth order derivatives are derived by using some accurate ansatz methods. 相似文献
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Three important nonlinear evolution equations are solved with the aid of the symbolic manipulation system.Maple,using the direct algebraic method proposed recently,We explicitly obtain several new solutions of physical interest in addition to rederiving all the known solutions. 相似文献
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Conservation laws for variable coefficient nonlinear wave equations with power nonlinearities
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Conservation laws for a class of variable coefficient nonlinear wave equations with power nonlinearities are investigated.The usual equivalence group and the generalized extended one including transformations which are nonlocal with respect to arbitrary elements are introduced.Then,using the most direct method,we carry out a classification of local conservation laws with characteristics of zero order for the equation under consideration up to equivalence relations generated by the generalized extended equivalence group.The equivalence with respect to this group and the correct choice of gauge coefficients of the equations play the major roles for simple and clear formulation of the final results. 相似文献
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A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model
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Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony (mBBM) model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations. 相似文献
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We have examined the influence of parametric noise on the solution behavioru(t, x) of a nonlinear initial value() problem arising in cell kinetics. In terms of ensemble statistics, the eventual limiting solution mean
and variance
are well-characterized functions of the noise statistics, and
and
depend on . When noise is continuously present along the trajectory,
and
are independent of the noise statistics and . However, in their evolution toward
and
, both
u
(t, x) and
u
2
(t, x) depend on the noise and. 相似文献