YARIM TO‘G‘RI CHIZIQDA RATSIONAL AKS ETTIRISH KOEFFITSIYENTLARIGA EGA MANFIY TARTIBLI KORTEVEG-DE FRIZ TENGLAMASINING ANIQ YECHIMLARINI MATRITSA-TRIPLETI KONSTRUKSIYASI ORQALI HOSIL QILISH

Urazbayev Gayrat Urazaliyevich  Babadjanova Aygul Komiljanovna   Eshchanova Gulchexra Sheribbayevna

Arxiv / №6 · 2026 / Maqola

YARIM TO‘G‘RI CHIZIQDA RATSIONAL AKS ETTIRISH KOEFFITSIYENTLARIGA EGA MANFIY TARTIBLI KORTEVEG-DE FRIZ TENGLAMASINING ANIQ YECHIMLARINI MATRITSA-TRIPLETI KONSTRUKSIYASI ORQALI HOSIL QILISH

Annotatsiya

O'ZBEKCHA

Ushbu maqolada yarim o‘qda berilgan manfiy tartibli Korteveg–de Friz sistemasi
, 4 2 0,txxxxxxxuvvuvuvu=−−+= ko‘rib chiqilib, Marchenko integral tenglamasi bilan bog‘liq matritsaviy uchlik usuli yordamida aniq ye-chimlar quriladi. Ratsional akslanish koeffitsiyentlari va diskret spektral ma’lumotlardan boshlab, chekli o‘lchamli (), , ABC matritsaviy uchlik quriladi hamda Marchenko yadrosi ();xtΓning ajraluvchi ko‘rinishi hosil qilinadi. Natijada teskari masala mat-ritsaviy tenglama ishtirokidagi algebraik matritsa sxemasiga keltiriladi. Maqolada uzluksiz spektrli ye-chimlar hamda diskret spektrga ega yechimlarni o‘z ichiga olgan bir nechta aniq misollar keltirilgan.

Kalit so'zlar: manfiy tartibli KdF tenglamasi, yarim o‘q masalasi, teskari sochilish usuli, Marchenko tenglamasi, uchlik matritsa, aniq yechimlar.

РУССКИЙ

В данной статье рассматривается система Кортевега-де Фриза отрицатель-ного порядка на полупрямой , 4 2 0,txxxxxxxuvvuvuvu=−−+=и строятся явные решения с помощью метода матричной тройки, связанного с интеграль-ным уравнением Марченко.Исходя из рациональных коэффициентов отражения и дискретных спектральных данных, строится ко-нечномерная матричная тройка (), , ABC и выводится сепа-рабельное представление ядра Марченко ();xtΓ Это позволяет свести обратную задачу к алгеб-раической матричной схеме, содержащей матричное уравнение. Приведены несколько явных примеров, включая решения с чисто непрерывным спектром и решения с дискретным спектром.

Ключевые слова: уравнение КдФ отрицательного порядка, задача на полупрямой, метод об-ратного рассеяния, уравнение Марченко, матричный триплет, явные решения.

ENGLISH

This paper considers the negative-order Korteweg–de Vries system on the halfline
, 4 2 0,txxxxxxxuvvuvuvu=−−+= and constructs explicit solutions by means of a matrix triplet method associated with the Marchenko integral equation. Starting from rational reflection coefficients and discrete spectral data, we construct a finite-dimensional triplet (), , ABC and derive a separable representation of the Marchenko kernel. This reduces the inverse problem to an algebraic matrix framework involving a matrix ();.xtΓ Several explicit examples are presented, including purely continuous-spectrum solutions and solutions with discrete spectrum.

Keywords: negative-order KdV equation, half-line problem, inverse scattering transform, Marchenko equation, matrix triplet, explicit solutions.

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Foydalanilgan adabiyotlar

Aktosun T. & Klaus M. (2001). Inverse theory: problem on the line. In: Pike, E.R. & Sabatier, P. C. (eds.) Scattering, pp. 770–785, London, Academic Press.
Aktosun T., Klaus M. & van der Mee, C. (1992). Explicit Wiener–Hopf factorization for certain nonrational matrix functions. Integral Equations and Operator Theory, 15, pp. 879–900.
Aktosun T. & van der Mee C. (2006). Explicit solutions to the Korteweg-de Vries equation on the half-line. Inverse Problems, 22, pp. 2165–2174.
Deift P. & Trubowitz E. (1979). Inverse scattering on the line. Communications on Pure and Applied Mathematics, 32, pp. 121–251.
Faddeev L.D. (1967). Properties of the S-matrix of the one-dimensional Schrödinger equation. American Mathematical Society Translations, 65, pp. 139–166.
Fokas A.S. (2002). Integrable nonlinear evolution equations on the half-line. Communications in Mathematical Physics, 230, pp. 1–39.
Gardner C S., Greene J.M., Kruskal M.D. & Miura R.M. (1967). Method for solving the Korteweg-de Vries equation. Physical Review Letters, 19, pp. 1095–1097.
Kriecherbauer T., Urazboev G.U. & Babadjanova A.K. (2025). On the matrix negative-order Korteweg-de Vries equation – the commutative case. Wave Motion, 138, 103562.
Marchenko V.A. (1986). Sturm–Liouville Operators and Applications. Basel: Birkhäuser.
Qiao Z. & Li J. (2011). Negative-order KdV equation with both solitons and kink wave solutions. Europhysics Letters, 94, 50003.

← Songa qaytish

[1] Aktosun T. & Klaus M. (2001). Inverse theory: problem on the line. In: Pike, E.R. & Sabatier, P. C. (eds.) Scattering, pp. 770–785, London, Academic Press.

[2] Aktosun T., Klaus M. & van der Mee, C. (1992). Explicit Wiener–Hopf factorization for certain nonrational matrix functions. Integral Equations and Operator Theory, 15, pp. 879–900.

[3] Aktosun T. & van der Mee C. (2006). Explicit solutions to the Korteweg-de Vries equation on the half-line. Inverse Problems, 22, pp. 2165–2174.

[4] Deift P. & Trubowitz E. (1979). Inverse scattering on the line. Communications on Pure and Applied Mathematics, 32, pp. 121–251.

[5] Faddeev L.D. (1967). Properties of the S-matrix of the one-dimensional Schrödinger equation. American Mathematical Society Translations, 65, pp. 139–166.

[6] Fokas A.S. (2002). Integrable nonlinear evolution equations on the half-line. Communications in Mathematical Physics, 230, pp. 1–39.

[7] Gardner C S., Greene J.M., Kruskal M.D. & Miura R.M. (1967). Method for solving the Korteweg-de Vries equation. Physical Review Letters, 19, pp. 1095–1097.

[8] Kriecherbauer T., Urazboev G.U. & Babadjanova A.K. (2025). On the matrix negative-order Korteweg-de Vries equation – the commutative case. Wave Motion, 138, 103562.

[9] Marchenko V.A. (1986). Sturm–Liouville Operators and Applications. Basel: Birkhäuser.

[10] Qiao Z. & Li J. (2011). Negative-order KdV equation with both solitons and kink wave solutions. Europhysics Letters, 94, 50003.