Summary) The abstract should follow the structure of the article (relevance, degree of exploration of the problem, the goal, the main results, conclusion) and characterize the theoretical and practical significance of the study results. The abstract should not contain wording echoing the title, cumbersome grammatical structures and abbreviations. The text should be written in scientific style. The volume of abstracts (summaries) depends on the content of the article, but should not be less than 250 words. All abbreviations must be disclosed in the summary (in spite of the fact that they will be disclosed in the main text of the article), references to the numbers of publications from reference list should not be made. The sentences of the abstract should constitute an integral text, which can be made by use of the words “consequently”, “for example”, “as a result”. Avoid the use of unnecessary introductory phrases (eg, “the author of the article considers...”, “The article presents...” and so on.)

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Ministry of Education and Science of the Russian Federation

Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

Solitons and the Inverse Scattering Transform

http://cds.cern.ch/record/398350/files/9909001.pdf

Gravitational wave experiments and early universe cosmology

TianQin is a proposal for a space-borne detector of gravitational waves in the millihertz frequencies. The experiment relies on a constellation of three dragfree spacecraft orbiting the Earth. Inter-spacecraft laser interferometry is used to monitor the distances between the test masses. The experiment is designed to be capable of detecting a signal with high confidence from a single source of gravitational waves within a few months of observing time. We describe the preliminary mission concept for TianQin, including the candidate source and experimental designs. We present estimates for the major constituents of the experiment's error budget and discuss the project's overall feasibility. Given the current level of technological readiness, we expect TianQin to be flown in the second half of the next decade.

https://iopscience.iop.org/article/10.1088/0264-9381/33/3/035010/pdf

TianQin: a space-borne gravitational wave detector

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

/pdf/backlund-and-darboux-transformations-geometry-and-modern-4k80wq4g1i.pdf

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

We present a Backlund transformation operating on solutions of the Korteweg-de Vries equation. It provides a unifying approach to the theory of solitary waves (solutions). Recursive application of the transformation to any solution of the Korteweg-de Vries equation generates a hierarchy of solutions for which an algebraic recursion relation is obtained. Every such hierarchy (one of which is the known family of pure multisoliton solutions) thus admits a nonlinear superposition principle.

Backlund transformation for solutions of the Korteweg-de Vries equation

The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrodinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open—i.e., does not terminate as a set of structure relations of a finite‐dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Backlund transformation, and derive the double Backlund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.

Prolongation structures of nonlinear evolution equations

A calculation is made of the effect of gravity waves on the observed Doppler shift of a sinusoidal electromagnetic signal transmitted to, and transponded from, a distant spacecraft. We find that the effect of plane gravity waves on such observations is not intuitively immediate and in fact can have surprisingly different spectral signatures for different spacecraft directions and distances. We suggest the possibility of detecting such plane waves by simultaneous coherent Doppler tracking of several spacecraft.

Response of Doppler spacecraft tracking to gravitational radiation

Ground-based, equal-arm-length laser interferometers are being built to measure high-frequency astrophysical graviatational waves. Because of the arm-length equality, laser light experiences the same delay in each arm and thus phase or frequency noise from the laser itself precisely cancels at the photodetector.

https://iopscience.iop.org/article/10.1086/308110/pdf

Time-delay interferometry for space-based gravitational wave searches

Methods are discussed for discovery of physically or mathematically special families of exact solutions of systems of partial differential equations. Such systems are described geometrically using equivalent sets of differential forms, and the theory derived for obtaining the generators of their invariance groups‐vector fields in the space of forms. These isovectors then lead naturally to all the special solutions discussed, and it appears that other special ansatze must similarly be capable of geometric description. Application is made to the one‐dimensional heat equation, the vacuum Maxwell equations, the Korteweg‐de Vries equation, one‐dimensional compressible fluid dynamics, the Lambropoulos equation, and the cylindrically symmetric Einstein‐Maxwell equations.

Frank B. Estabrook

Papers

Backlund transformation for solutions of the Korteweg-de Vries equation

Prolongation structures of nonlinear evolution equations

Response of Doppler spacecraft tracking to gravitational radiation

Time-delay interferometry for space-based gravitational wave searches

Geometric Approach to Invariance Groups and Solution of Partial Differential Systems