Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types

Lü, Xing; Chen, Si-Jia

doi:10.1007/s11071-020-06068-6

Cited by 247 publications

(55 citation statements)

References 52 publications

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“…There should exist more interesting problems which need to be discussed. We expect to investigate integrable properties 35,36 and high-order lump solutions [37][38][39][40][41][42][43][44] to the presented Hirota bilinear equations in our future works.…”

Section: Discussionmentioning

confidence: 99%

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Cheng

2021

Math Methods in App Sciences

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A universal property in terms of Hirota differential operators is presented, with an aim to construct Hirota bilinear equations possessing Pfaffian formulations in (2+1)-dimensions. It is then shown that there exist linear combination solutions of exponential waves to various extended Hirota bilinear equations. Applications are made for the (2+1)-dimensional generalized Hirota-Satsuma-Ito (HSI) equation, the special fourth-order (2+1)-dimensional nonlinear equation, and the fifth-order Korteweg-de Vries (KdV) equation, thereby presenting their particular Pfaffian and multiple wave solutions.

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Section: Discussionmentioning

confidence: 99%

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Cheng

2021

Math Methods in App Sciences

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“…With the advent of digital cameras and other smart devices, a large number of tasks that should be done by humans are transferred to machines for automation development, and with the advancement of deep learning, image restoration based on deep learning is gradually growing. The combination of sparse coding and deep neural networks with self-encoders was proposed in the literature, which is more effective compared to traditional image restoration algorithms; then, two different multilayer perceptronbased image restoration algorithms were proposed, but both of them are generally effective in restoring larger blocks of occluded areas; contextual content-based image restoration algorithms were proposed in the literature, where highdimensional features of the image are learned using a selfcoding network and detail content repair uses adversarial network, but the method is difficult to train; DCGAN deep network-based image semantic repair algorithm generating model is proposed in the literature to repair the mutilated regions by iteratively training and optimizing the network model; the disadvantages of this algorithm include long training time, easy gradient explosion, and cannot be applied in real-time [4].…”

Section: Related Workmentioning

confidence: 99%

“…The multinetwork strategy does have an efficiency advantage over the single-network strategy, but it does not mean that the multinetwork strategy is superior in all cases. 4 Advances in Mathematical Physics…”

Section: Partial Differential Equations Based On Imagementioning

confidence: 99%

A Partial Differential Equation-Based Image Restoration Method in Environmental Art Design

2021

Advances in Mathematical Physics

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With the rapid development of networks and the emergence of various devices, images have become the main form of information transmission in real life. Image restoration, as an important branch of image processing, can be applied to real-life situations such as pixel loss in image transmission or network prone to packet loss. However, existing image restoration algorithms have disadvantages such as fuzzy restoration effect and slow speed; to solve such problems, this paper adopts a dual discriminator model based on generative adversarial networks, which effectively improves the restoration accuracy by adding local discriminators to track the information of local missing regions of images. However, the model is not optimistic in generating reasonable semantic information, and for this reason, a partial differential equation-based image restoration model is proposed. A classifier and a feature extraction network are added to the dual discriminator model to provide category, style, and content loss constraints to the generative network, respectively. To address the training instability problem of discriminator design, spectral normalization is introduced to the discriminator design. Extensive experiments are conducted on a data dataset of partial differential equations, and the results show that the partial differential equation-based image restoration model provides significant improvements in image restoration over previous methods and that image restoration techniques are exceptionally important in the application of environmental art design.

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“…The literature [6] converts the multifocus retrieval task into a spot and scatter focus region boundary detection task starting from boundary segmentation, where the source image focus information is segmented out by the detected boundaries of different focus regions, and the regions with different degrees of focus segmented from each source image will be used as the respective retrieval decision maps. For the decision map of multisource images, the region with a higher focus degree is used as a high priority decision to guide the multifocused images to complete the retrieval task.…”

Section: Related Workmentioning

confidence: 99%

Multifeatured Image Retrieval Techniques Based on Partial Differential Equations for Online Shopping

2021

Advances in Mathematical Physics

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In today’s rapid development of network and multimedia technology, the booming of electronic commerce, users in the network shopping species of images and other multimedia information showing geometric growth, in the face of this situation, how to find the images they need in the vast amount of online shopping images has become an urgent problem to solve. This paper is based on the partial differential equation to do the following research: Based on the partial differential equation is a kind of equation that simulates the human visual perception system to analyze images; based on the summary of the advantages and disadvantages of multifeature image retrieval technology, we propose a multifeature image retrieval technology method based on the partial differential equation to alleviate the indexing imbalance caused by the mismatch of multifeature image retrieval technology distribution. To improve the search speed of the data-dependent locally sensitive hashing algorithm, we propose a query pruning algorithm compatible with the proposed partial differential equation-based multifeature image retrieval technology method, which greatly improves the retrieval speed while ensuring the retrieval accuracy; to implement the data-dependent partial differential equation algorithm, we need to distribute the data set among different operation nodes, and to better achieve better parallelization of operations, we need to measure the similarity between categories, and we achieve the problem of distributing data among various categories in each operation node by introducing a clustering method with constraints. The purpose of this article for image recognition is for better shopping platforms for merchants. This algorithm has trained multiple samples and has data support. The experimental results show that our proposed data set allocation method shows significant advantages over the data set allocation method that does not consider category correlation. However, the image features used in image retrieval systems are often hundreds or even thousands of dimensions, and these features are not only high in dimensionality but also huge in number, which makes image retrieval systems encounter an inevitable problem—“dimensionality disaster.” To overcome this problem, scholars have proposed a series of approximate nearest neighbor methods, but multifeature image retrieval techniques based on partial differential equations are more widely used in people’s daily life.

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Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types

Cited by 247 publications

References 52 publications

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

A Partial Differential Equation-Based Image Restoration Method in Environmental Art Design

Multifeatured Image Retrieval Techniques Based on Partial Differential Equations for Online Shopping

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