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Solitary wave solutions to nonlinear partial differential equations (PDE) play a major role in mathematical physics. A huge rise of interest to solitary wave solutions has been observed during the last decade. That can be explained by the immense rise of the computational power available and applicable for symbolic computations and transformations. Symbolic computations have been extensively used as a powerful mathematical tool for solving high-dimensional nonlinear evolutions in mathematical physics.

A number of methods for the identification of solitary wave solutions have been developed during the last decades. Homogeneous balance method, the Exp-function method, the tanh method and its various extensions, the (G’/G) expansion method, the auxiliary (or subsidiary) differential equation method, and many other similar methods are successfully used to seek new solitary wave solutions to nonlinear PDE. The key idea of most of these methods is that the traveling wave solution of a complicated nonlinear evolution equation can be guessed (supposed) as a ratio of polynomials of standard functions whose argument is a traveling wave term. The degree of the polynomial can be determined by the homogeneous (direct) balance between the highest derivatives and the nonlinear terms in the original PDE. However, a straightforward application of these methods has attracted a considerable amount of criticism

The main criticism addressed to these methods is based on one of the two important problems. First of all, it has been demonstrated that methods based on the direct balancing technique can produce solutions which do not satisfy the original differential equation. Secondly, it has been demonstrated that methods based on the direct balancing technique cannot produce the necessary and sufficient conditions for the existence of solitary wave solutions neither in the space of system parameters, nor in the space of initial conditions.

A special attention in this Special Issue is also focused to fractional PDE. Many techniques for the construction of solitary wave solutions to fractional PDE did attract a lot of criticism during the last decade, too. For example, a primitive travelling wave variable substitution cannot be used to transform a fractional PDE into an ODE without a careful analysis of the existence conditions.

The main focus of this Special Issue is concentrated to mathematical methods and techniques which can be employed to construct solitary wave solutions to nonlinear PDE (including fractional PDE), and which do not possess the mentioned drawbacks. The main goal of the Special Issue is not to derive new solitary wave solutions to nonlinear evolutions. On the contrary, the main objective of this Special Issue is to reflect the latest state of the art in contemporary methods and techniques for the construction of solitary wave solutions to PDE and fractional PDE.

The main topics of this Special Issue:
• Modern mathematical techniques for the construction of solitary wave solutions to PDE
• Modern mathematical techniques for the construction of solitary wave solutions to fractional PDE

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