On symmetries of KdV-like evolution equations

The $x$-dependence of the symmetries of (1+1)-dimensional scalar translationally invariant evolution equations is described. The sufficient condition of (quasi)polynomiality in time $t$ of the symmetries of evolution equations with constant separant is found. The general form of time dependence of the symmetries of KdV-like non-linearizable evolution equations is presented.Comment: LaTeX, 8 pages, no figures, very minor change

Group classification of variable coefficient KdV-like equations

The exhaustive group classification of the class of KdV-like equations with time-dependent coefficients $u_t+uu_x+g(t)u_{xxx}+h(t)u=0$ is carried out using equivalence based approach. A simple way for the construction of exact solutions of KdV-like equations using equivalence transformations is described.Comment: 8 pages; minor misprints are corrected. arXiv admin note: substantial text overlap with arXiv:1104.198

Conservation laws and normal forms of evolution equations

We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.Comment: 16 page

French diplomats and the military on Soviet Russia and the balance of power in Central-Eastern Europe in 1922

The consolidation of the Soviet state in 1922 and the activities of Soviet diplomacy in the key international forums had a direct impact on the strategic situation in Europe. The eventual strengthening of Soviet Russia/the USSR was both a threat and an opportunity for France as one of the leading European powers of that period, which had obligations and interests in Central and Eastern Europe. The author aims to identify the main approaches of French diplomats and the military to a set of issues related to the possible development of Soviet Russia in 1922 and its place in the European balance of power. The study is based on a wide range of primary sources from the Diplomatic Archives of the Ministry of Europe and Foreign Affairs of France, the National Archives of France, the Historical Service of the Ministry of the Armed Forces of the Fifth Republic, as well as on recently published French diplomatic and military documents. The author concludes that the French elites had a rather ambiguous attitude towards the process and the first results of political consolidation and socio-economic development of the Soviet state. On the one hand, the formation of the USSR was an obvious manifestation of the growing Soviet power that somewhat diminished the hopes of French officials for the imminent fall of the Bolsheviks. At the same time, diplomats and the military both in Paris and on-site were often skeptical about the prospects for the development of the Soviet economy, noting the catastrophic consequences of hunger, economic and financial ruin. Moderate optimism about the opportunity to intensify trade and economic contacts with Soviet Russia as its economy recovers coexisted with pronounced pessimism. The French assessments of the military potential of the Soviet state were marked by the same ambivalence. The acknowledgement of the current limited capabilities of the Red Army and the Red Fleet was accompanied by the growing recognition that the basis of the military power of the Soviet state had not been undermined. All this could help Moscow improve its international stance in the future, which would inevitably affect the balance of power in Europe. Under these circumstances, the French elites debated the prospects for the ‘normalization’ of the Bolshevik regime and its incorporation into the Versailles order. The author argues that all these contradictory attitudes, views and assessments that surfaced in 1922 to a large extent predetermined the overall direction and specific content of the French policy towards the USSR in the following years

Did anyone believe in the Lea gue of Nations ? The international or gani zation and the forei gn policies of France and Great Britain in the 1920s

В статье рассматривается вопрос о роли и месте Лиги Наций во внешнеполитических курсах Франции и Великобритании в 1920-е гг. Выделены функции международной организации как инструмента коллективной безопасности, сокращения вооружений, разрешения локальных конфликтов, снижения напряженности в преддверии вероятной эскалации конфликта. Автор пришел к выводу о том, что в Париже и Лондоне испытывали скепсис по поводу эффективности Лиги Наций как инструмента коллективной безопасности и разоружения, хотя и отмечали потенциальную позитивную роль организации в двух других ее «ипостасях».This article explores the role played by the League of Nations in the foreign policies of France and Great Britain during the 1920s. International organisation is analysed as the instrument to achieve the collective security, to reduce the armaments, to resolute the local conflicts, and, finally, to lessen international tensions and to prevent them from escalating to war. Author concludes that Paris and London were skeptical about the League of Nations as the instrument of collective security and disarmament, though they appreciated two its other functions.Исследование выполнено при финансовой поддержке РФФИ, проект «От интервенции к признанию: “русская” политика Французской Республики в 1917–1924 гг.» №18-09-00498 А

Local conservation laws of second-order evolution equations

Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.Comment: 11 pages, minor correction