Dispersionless equation
Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system. Examples. Dispersionless KP equation. The dispersionless Kadomtsev–Petviashvili equation (dKPE), also known (up to an inessential linear change of variables) as the Khokhlov–Zabolotskaya equation, has the form It arises from the commutation of the following pair of 1-parameter families of vector fields where formula_5 is a spectral parameter. The dKPE  is the formula_6-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation, arising when considering long waves of that system. The dKPE, like many other (2+1)-dimensional integrable dispersionless systems, admits a (3+1)-dimensional generalization. The Benney moment equations. The dispersionless KP system is closely related to the Benney moment hierarchy, each of which is a dispersionless integrable system: These arise as the consistency condition between and the simplest two evolutions in the hierarchy are: The dKP is recovered on setting and eliminating the other moments, as well as identifying formula_12 and formula_13. If one sets formula_14, so that  the countably many moments formula_15 are expressed in terms of just two functions, the classical shallow water equations result: These may also be derived from considering slowly modulated wave train solutions of the nonlinear Schrodinger equation. Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the Gibbons-Tsarev equation. Dispersionless Korteweg–de Vries equation. The dispersionless Korteweg–de Vries equation (dKdVE) reads as It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation. It is satisfied by formula_19-independent solutions of the dKP system. It is also obtainable from the formula_20 -flow of the Benney hierarchy on setting Dispersionless Novikov–Veselov equation. The dispersionless Novikov-Veselov equation is most commonly written as the following equation for a real-valued function formula_22: where the following standard notation of complex analysis is used: formula_24, formula_25. The function formula_26 here is an auxiliary function, defined uniquely from formula_27 up to a holomorphic summand. Multidimensional integrable dispersionless systems. See for systems with contact Lax pairs, and e.g., and references therein for other systems. https://rikkyo.repo.nii.ac.jp/index.php?action=pages_view_main&active_action=repository_action_common_download&item_id=9046&item_no=1&attribute_id=22&file_no=1&page_id=13&block_id=49