Interaction of kinks for semilinear wave equations with a small parameter

Shock wave formation process for a multidimensional scalar conservation law

“…For instance, using this method, we are able to find explicit formulas describing the interaction of solitons in the case of generalized KdV equations [5,10], the interaction of Sine-Gordon solitons [14,21], the evolution of nonlinear waves in the case of scalar conservation laws [3,7], the interaction [6,11] and formation [8,25] of δ-shock waves in the case of a triangular system of conservation laws, δ -shock waves as a new type of singular solution of hyperbolic systems of conservation laws [27], the confluence of free boundaries in the Stefan problem with underheating [4], and different interactions of the shock waves appearing on the gas dynamics [12,13,15], etc. In the beginning, we introduce definitions and the fundamental theorem of the method that we are going to use: the weak asymptotic method.…”

Soliton‐type asymptotics for non‐integrable equations: a survey

“…Originally, such idea had been suggested by V. Danilov and V. Shelkovich for shock wave type solutions (1997, [15]), and after that, it has been developed and adapted for many other problems (V. Danilov, G. Omel'yanov, V. Shelkovich, D. Mitrovic and others, [14,[16][17][18][19][20][21][22][23][24] and references therein). We called this approach the 'weak asymptotics method' .…”

“…Because ı.x '/ and ı 0 .x '/ are linearly independent, their coefficients in (34) should be equal to 0. Taking into account the identities (3) and (7), we obtain again the Equations (15) and (19) for the one-phase asymptotics (12). Let us revert to the problem (1), (6) of two-wave interaction.…”

“…Reverting to the single-phase asymptotic solution (12), one can easily establish the orthogonality condition (17), and therefore the Equation (19), is the integral form of (54), calculated for (12) with the accuracy O D 0 ." 2 /.…”

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“…The method is intensively used in recent years for investigations of nonlinear wave phenomena. For instance, using this method, we are able to find explicit formulas describing the interaction of solitons in the case of generalized KdV equations [5,10], the interaction of Sine-Gordon solitons [14,21], the evolution of nonlinear waves in the case of scalar conservation laws [3,7], the interaction [6,11] and formation [8,25] of δ-shock waves in the case of a triangular system of conservation laws, δ -shock waves as a new type of singular solution of hyperbolic systems of conservation laws [27], the confluence of free boundaries in the Stefan problem with underheating [4], and different interactions of the shock waves appearing on the gas dynamics [12,13,15], etc.…”